Simon Lauret

Elise Noel-Savina

Grégoire Prévot

M.D Nicolas Guibert

Ph.D Laurent Lionel Reber

Danièle Brouquières

M.D Alain Didier

M.D Laurent Guilleminault

Are serum immunoglobulin concentrations a predictive biomarker of response to anti-IL5/IL5Rα therapies?

Keywords: interleukin 5: IL5, interleukin 5 receptor: IL5R, ROC: receiver operating characteristic asthma, mepolizumab, benralizumab, IgG, biomarkers

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Introduction

Severe asthma is a global health problem that affects approximatively 3.6% of all asthma patients [START_REF] Hekking | The prevalence of severe refractory asthma[END_REF]. Severe asthma is defined as asthma requiring a high level of treatment [START_REF] Chung | International ERS/ATS guidelines on definition, evaluation and treatment of severe asthma[END_REF]. Despite this treatment, some patients with severe asthma experience frequent exacerbations and hospitalizations [START_REF] Hossny | Severe asthma and quality of life[END_REF][START_REF] Lang | Severe asthma: epidemiology, burden of illness, and heterogeneity[END_REF]. Severe asthma is also associated with a high economic burden and gathers the major part of healthcare costs among these patients [START_REF] Nordon | Clinical and economic burden of severe asthma: A French cohort study[END_REF].

Severe asthma is a heterogeneous condition including many different phenotypes [START_REF] Wenzel | Asthma phenotypes: the evolution from clinical to molecular approaches[END_REF]. A severe eosinophilic asthma phenotype has been previously identified in several cluster-based studies [START_REF] Haldar | Cluster analysis and clinical asthma phenotypes[END_REF][START_REF] Moore | Identification of asthma phenotypes using cluster analysis in the Severe Asthma Research Program[END_REF] and it is characterized by severe asthma with type 2-high inflammation [START_REF] Amelink | Severe adult-onset asthma: a distinct phenotype[END_REF]. Recently, biological therapies targeting eosinophils being the main components of T2 inflammation have been approved [START_REF] Keating | Mepolizumab: first global approval[END_REF][START_REF] Markham | Benralizumab: first global approval[END_REF]. Mepolizumab, an anti-interleukin-5 monoclonal antibody (anti-IL5), and benralizumab, an anti-IL5 receptor-α (IL5Rα) antibody, are used as add-on therapies for patients with severe uncontrolled eosinophilic asthma (blood eosinophils count ≥ 0.3 G/l). Mepolizumab and benralizumab are associated with oral corticosteroid sparing as well as a significant reduction in the rate of exacerbations [START_REF] Ortega | Severe eosinophilic asthma treated with mepolizumab stratified by baseline eosinophil thresholds: a secondary analysis of the DREAM and MENSA studies[END_REF][START_REF] Bleecker | Efficacy and safety of benralizumab for patients with severe asthma uncontrolled with high-dosage inhaled corticosteroids and long-acting β2-agonists (SIROCCO): a randomised, multicentre, placebo-controlled phase 3 trial[END_REF]. Although the response rate of those therapies is very high, some patients show no response to anti-IL5/IL5Rα therapies. In BORA study, an extension of phase III clinical trials, 28% of 527 patients who received benralizumab still experienced at least one exacerbation by week 56 [START_REF] Busse | Long-term safety and efficacy of benralizumab in patients with severe, uncontrolled asthma: 1-year results from the BORA phase 3 extension trial[END_REF]. In MENSA, phase III study on mepolizumab, 38 % of patients experienced no significant improvement with mepolizumab [START_REF] Ortega | Mepolizumab treatment in patients with severe eosinophilic asthma[END_REF].

In clinical practice, patient's selection before the beginning of biological therapies has been very challenging because none predictive biomarkers have been identified except blood eosinophil count. Despite this latter, clinicians are still unable to accurately predict the response to these therapies. Therefore, predictive biomarkers of anti-IL5/IL5Rα therapies are urgently needed [START_REF] Yancey | Biomarkers for severe eosinophilic asthma[END_REF].

Immunoglobulins (Igs) are circulating heterodimeric proteins primarily involved in adaptive immunity [START_REF] Schroeder | Structure and function of immunoglobulins[END_REF]. In asthma, it has been established that a low level of serum IgG is associated with recurrent exacerbations [START_REF] Lee | Association between primary immunodeficiency and asthma exacerbation in adult asthmatics[END_REF]. Moreover, the serum IgG level is significantly lower when asthma exacerbations are associated with positive viral samples [START_REF] Verduyn | Serum IgG Concentrations in Adult Patients Experiencing Virus-Induced Severe Asthma Exacerbations. The journal of allergy and clinical immunology[END_REF]. Asthma patients with hypogammaglobulinemia have low type-2 inflammation markers [START_REF] Dupin | Asthma and Hypogammaglobulinemia: an Asthma Phenotype with Low Type 2 Inflammation[END_REF]. According to this, it would be interesting to assess serum IgG concentrations as a biomarker of anti-IL5/IL5Rα therapies in severe asthma.

The aim of our study was to determine whether baseline serum IgG concentrations are a predictive biomarker of anti-IL5/IL5Rα therapies.

Methods

Study design

A retrospective "real-life" study was carried out on severe asthma patients between March 2018 and October 2019 at Toulouse University Hospital Center. The patients were included if they had a quantification of serum Ig concentration at baseline and after a 6-month treatment of mepolizumab or benralizumab prescribed for severe asthma. Severe asthma was defined according to the ATS/ERS guidelines [START_REF] Holguin | Management of severe asthma: a European Respiratory Society/American Thoracic Society guideline[END_REF]. Benralizumab or mepolizumab were administered according to drug manufacturers' recommendations. Considering our study as a "real life" one, the prescription of mepolizumab or benralizumab was at the discretion of the clinicians.

Regarding the study design, severe asthma patients had serum immunoglobulin quantification at baseline before the beginning of mepolizumab or benralizumab. After a 6-month treatment of mepolizumab or benralizumab, patients received a second serum immunoglobulin quantification.

The database was anonymous and compliant with the restrictive requirements of the National Commission for Data Protection (CNIL number: 2206723 v 0), the body that oversees the enforcement of data privacy laws in France. Our study was conducted in accordance with the French law. Moreover, the study protocol was approved by the Institutional Review Board of the French Learned Society for Respiratory Medicine (Société de Pneumologie de Langue Française) (CEPRO 2020-042). In accordance with French law, information was provided to the patients and oral non-opposition was collected.

Clinical data and response to biological therapies

Age, gender, BMI, smoking history, the annualized rate of asthma exacerbations, asthma control test (ACT) score, the presence of nasal polyposis, positive prick tests to aeroallergens, the daily dose of inhaled corticosteroids and the use of long-term oral corticosteroids (defined by systemic corticosteroids for more than 6 months) were collected.

The response to mepolizumab or benralizumab was evaluated at 6 months using the GETE score (global evaluation of treatment effectiveness) [START_REF] Bousquet | Predicting and evaluating response to omalizumab in patients with severe allergic asthma[END_REF]. Patients with Excellent or good response were classified as responders, while those with a GETE score rated as moderate, poor, or worsening response were classified as non-responders.

Serum immunoglobulins

The concentration of serum Igs was collected at the initiation of mepolizumab or benralizumab and at 6 months of treatment. Serum Igs were quantified by turbidimetry and expressed in g/l.

IgE were quantified by an immunoenzymofluorimetric technique and expressed in kIU/l.

Lung function tests

Lung function tests were carried out using a spirometer (Jaeger Vyntus, CareFusion, Germany).

Values for Forced Vital Capacity (FVC) and Forced Expiratory Volume in one second (FEV1) were expressed in percentages of mean predicted values according to gender, height and age.

Measurements were performed in accordance with ATS/ERS guidelines [START_REF] Laszlo | Standardisation of lung function testing: helpful guidance from the ATS/ERS Task Force[END_REF][START_REF] Thoracic | ATS/ERS recommendations for standardized procedures for the online and offline measurement of exhaled lower respiratory nitric oxide and nasal nitric oxide, 2005[END_REF]. The mean predicted values were those of ERS for FEV1 and FVC [START_REF] Quanjer | Lung volumes and forced ventilatory flows[END_REF]. Fractional exhaled nitric oxide (FeNO) were obtained using an electrochemical device (Hypair FeNO, Medisoft®, Belgium).

Briefly, repeated and reproducible exhalations were performed, until obtaining at least two nitric oxide (NO) plateau values differing by less than 10%. Flow rate exhalation was 50 ml/sec. The retained FeNO value was the mean of two reproducible measurements and was expressed in parts per billions (ppb). FeNO was measured according to ATS/ERS guidelines [START_REF] Thoracic | ATS/ERS recommendations for standardized procedures for the online and offline measurement of exhaled lower respiratory nitric oxide and nasal nitric oxide, 2005[END_REF].

Statistical analysis

Quantitative data are expressed as median and [interquartile range] whereas qualitative data are expressed as number of patients (%). The Wilcoxon test is used to compare two quantitative variables. Statistical significance is defined as a p value of <0.05.

Receiver operator characteristics (ROC) curves are used to assess blood eosinophil count or serum IgG concentrations as potential predictive biomarkers of response to mepolizumab or benralizumab. Sensitivity and specificity of IgG cut-off for predicting responders to mepolizumab or benralizumab are also defined. The best IgG cut-off is determined using Youden's index.

Results

A total of 50 severe asthma patients treated with mepolizumab or benralizumab were included.

The details of patients' characteristics at baseline are indicated in Table 1. Median age was 56.0 At 6 months, the ACT score significantly improved with mepolizumab or benralizumab (median ACT score was 13 [START_REF] Amelink | Severe adult-onset asthma: a distinct phenotype[END_REF][START_REF] Keating | Mepolizumab: first global approval[END_REF][START_REF] Markham | Benralizumab: first global approval[END_REF][START_REF] Ortega | Severe eosinophilic asthma treated with mepolizumab stratified by baseline eosinophil thresholds: a secondary analysis of the DREAM and MENSA studies[END_REF][START_REF] Bleecker | Efficacy and safety of benralizumab for patients with severe asthma uncontrolled with high-dosage inhaled corticosteroids and long-acting β2-agonists (SIROCCO): a randomised, multicentre, placebo-controlled phase 3 trial[END_REF][START_REF] Busse | Long-term safety and efficacy of benralizumab in patients with severe, uncontrolled asthma: 1-year results from the BORA phase 3 extension trial[END_REF][START_REF] Ortega | Mepolizumab treatment in patients with severe eosinophilic asthma[END_REF][START_REF] Yancey | Biomarkers for severe eosinophilic asthma[END_REF] at baseline and 21 [START_REF] Bleecker | Efficacy and safety of benralizumab for patients with severe asthma uncontrolled with high-dosage inhaled corticosteroids and long-acting β2-agonists (SIROCCO): a randomised, multicentre, placebo-controlled phase 3 trial[END_REF][START_REF] Busse | Long-term safety and efficacy of benralizumab in patients with severe, uncontrolled asthma: 1-year results from the BORA phase 3 extension trial[END_REF][START_REF] Ortega | Mepolizumab treatment in patients with severe eosinophilic asthma[END_REF][START_REF] Yancey | Biomarkers for severe eosinophilic asthma[END_REF][START_REF] Schroeder | Structure and function of immunoglobulins[END_REF][START_REF] Lee | Association between primary immunodeficiency and asthma exacerbation in adult asthmatics[END_REF][START_REF] Verduyn | Serum IgG Concentrations in Adult Patients Experiencing Virus-Induced Severe Asthma Exacerbations. The journal of allergy and clinical immunology[END_REF][START_REF] Dupin | Asthma and Hypogammaglobulinemia: an Asthma Phenotype with Low Type 2 Inflammation[END_REF][START_REF] Holguin | Management of severe asthma: a European Respiratory Society/American Thoracic Society guideline[END_REF][START_REF] Bousquet | Predicting and evaluating response to omalizumab in patients with severe allergic asthma[END_REF][START_REF] Laszlo | Standardisation of lung function testing: helpful guidance from the ATS/ERS Task Force[END_REF] at 6 months; p<0.0001) (Table 2). A significant reduction of the annualized rate of exacerbations was also observed after a 6-month treatment of mepolizumab or benralizumab (6.0 [4.0-11.5] at baseline vs 0.0 [0.0-2.0] at 6 months; p<0.0001). In patients treated with a long-term use of oral steroids, the daily dose of OCS was significantly reduced with mepolizumab or benralizumab at 6 months ( 17 Regarding the change over time of serum immunoglobulin concentrations after a 6-month treatment of mepolizumab or benralizumab, IgA and IgM were not influenced by those treatments. However, IgG significantly increased from 9.2 [7.8-10.2] g/l at baseline to 10.1 [8.38-11.1] g/l at 6 months (p=0.001) (Figure 1). The long-term use of oral steroids seems to have no impact on the change in serum Ig concentrations over time. Actually, at 6 months, a similar increase in serum IgG concentrations was observed between patients treated with oral steroids and those with no oral steroids at baseline (+10.1 [-4.9-+22.0]% and +8.2 [+0.1-+17.5]%, respectively; p=0.93) (Table 3). However, among patients with OCS at baseline, we compare those with a 50% decrease in OCS dose and those with a decrease less than 50% at 6 months. There is a statistical difference between groups (+18.3 [+12.4-+26.6]% and -3.4 [-8.7-+0.2]%, respectively (p=0.0009). Regarding the concentrations of other serum Igs, the change over time was similar in patients with oral steroids and those with no oral steroids at baseline.

According to ROC curves, we determined whether serum IgG concentrations at baseline may be a good predictive biomarker of a response to mepolizumab or benralizumab (Figure 2). The area under the ROC curves for predicting the treatment response was 0.58 [95%IC 0.40-0.77] for blood eosinophil count (p=0.37) and 0.75 [95%IC: 0.58-0.92] for serum IgG concentrations (p=0.009). According to Youden's index, the best sensitivity and specificity were 69.2% and 62.9%, 76.9% and 75.7%, respectively for blood eosinophil count (0.48 G/l cut-off) and serum IgG concentrations (9.2 g/l cut off).

Discussion

A significant increase in serum IgG concentrations was observed 6 months after the beginning of anti-IL5/IL5Rα therapies. Moreover, sensitivity and specificity were determined for baseline serum IgG concentrations as a predictor of a good response to mepolizumab or benralizumab (76.9% and 75.7%, respectively for a 9.2 g/l cut-off).

Eosinophils are innate immune cells originated in pluripotent progenitor cells in the bone marrow and their survival, expansion, and terminal differentiation are mainly driven by IL5 [START_REF] Lee | Human versus mouse eosinophils: "that which we call an eosinophil, by any other name would stain as red[END_REF].

The physiological role of eosinophils has not been fully elucidated yet. However, their involvement in asthma pathophysiology (airway dysfunction and tissue remodelling) has been clearly established [START_REF] Rosenberg | Eosinophils: changing perspectives in health and disease[END_REF]. The beneficial effect of anti-IL5/IL5Rα therapies on asthma outcomes in severe eosinophilic asthma patients has been demonstrated but the mechanism involved has not been fully understood yet [START_REF] Pelaia | Benralizumab: From the Basic Mechanism of Action to the Potential Use in the Biological Therapy of Severe Eosinophilic Asthma[END_REF]. Eosinophils are more multifunctional than originally shown and these cells interact with several other immune cells such as T cells, B cells, macrophages or dendritic cells. For example, eosinophils release cytokines that influence T cell differentiation and also secrete chemoattractants for the recruitment of T cells, macrophages, and dendritic cells to tissue sites [START_REF] Spencer | Human eosinophils constitutively express multiple Th1, Th2, and immunoregulatory cytokines that are secreted rapidly and differentially[END_REF][START_REF] Akuthota | Immunoregulatory roles of eosinophils: a new look at a familiar cell[END_REF]. The effect of eosinophils on immunoglobulin production has not been extensively studied. Interestingly, it has been shown that bone marrow eosinophils are the key providers of plasma cell survival factors in mice [START_REF] Chu | Eosinophils are required for the maintenance of plasma cells in the bone marrow[END_REF]. Eosinophils are crucial for the retention of plasma cells in the bone marrow [START_REF] Chu | Eosinophils are required for the maintenance of plasma cells in the bone marrow[END_REF][START_REF] Chu | Immunization induces activation of bone marrow eosinophils required for plasma cell survival[END_REF]. Regarding the increase in serum IgG concentrations in our study, we hypothesise that mepolizumab or benralizumab may lead to plasma cell release into the bloodstream with eosinophil inhibition. However, it has also been shown that eosinophils promote B cell survival, proliferation, and immunoglobulin secretion in in vitro studies [START_REF] Wong | Eosinophils regulate peripheral B cell numbers in both mice and humans[END_REF]. A positive correlation between blood eosinophils and B cells has also been observed in hypereosinophilic patients [START_REF] Wong | Eosinophils regulate peripheral B cell numbers in both mice and humans[END_REF]. The interaction between eosinophils and the immune system is probably more complex and further studies are warranted [START_REF] Berek | Eosinophils: important players in humoral immunity[END_REF].

Systemic steroids may play a role in serum IgG increase within biological therapies. In fact, it is well known that anti-IL5/IL5Rα therapies are associated with a reduction of OCS use. In addition, steroids impact immunoglobulin production. Indeed, a significant reduction of serum IgG level has been observed with short-term [START_REF] Settipane | Corticosteroid effect on immunoglobulins[END_REF][START_REF] Mcmillan | The effect of corticosteroids on human IgG synthesis[END_REF] or long-term [37] use of OCS. There was no influence in IgA or IgM. In our study, a growth in serum IgG was particularly found at 6 months after the initiation of biological therapies in patients who had a significant reduction of OCS (long-term users). However, no difference in terms of IgG increase was observed between patients with or without long-term use of OCS at baseline. Therefore, we hypothesize that even in patients with no long-term use of OCS, a reduction of short-term courses of OCS with biological therapies may have an impact on serum IgG. Moreover, no change was observed in other immunoglobulins, given IgG are more sensitive to OCS than IgA or IgM.

Another explanation may be an artificial increase in serum IgG since patients were given injections of mepolizumab or benralizumab which are IgG1. The low amount of drugs administered to these patients cannot lead to such an increase in serum IgG. Moreover, no increase in IgG1 was observed. Immune complexes between anti-drug antibodies and mepolizumab or benralizumab have also been described in clinical trials. Given the method of serum IgG quantification, an artificial increase in serum IgG may be explained by immune complexes. However, this phenomenon is uncommon and does not explain such an increase [START_REF] Liao | Inhibition of interleukin-5 induced false positive anti-drug antibody responses against mepolizumab through the use of a competitive blocking antibody[END_REF].

The usefulness of serum IgG in severe asthma management is still controversial. The prevalence of hypogammaglobulinemia seems to be higher in asthma patients compared to the general population [START_REF] Hamilos | Hypogammaglobulinemia in asthmatic patients[END_REF]. In this study, hypogammaglobulinemia was strongly associated with the use of systemic corticosteroids. Asthma patients with hypogammaglobulinemia can be integrated in a specific phenotype with low type-2 inflammation [START_REF] Dupin | Asthma and Hypogammaglobulinemia: an Asthma Phenotype with Low Type 2 Inflammation[END_REF]. In primary immune deficiency, low serum immunoglobulin concentrations expose asthma patients to a higher risk of asthma exacerbations and severe asthma. In fact, it has been estimated that the relative risk of asthma exacerbation was 1.70 times higher in patients with low serum immunoglobulin concentrations (low IgG, A, M or IgG subclasses) compared to those with normal concentrations [START_REF] Lee | Association between primary immunodeficiency and asthma exacerbation in adult asthmatics[END_REF].

Moreover, severe asthma prevalence also appears to be significantly higher in patients with low serum immunoglobulin concentrations compared to those with normal concentrations (32.5% and 13.0%, respectively) [START_REF] Lee | Association between primary immunodeficiency and asthma exacerbation in adult asthmatics[END_REF]. As observed in patients with common variable deficiency, it has been suggested that the recurrence of viral or bacterial infections is a mechanism of asthma exacerbations in patients with low serum IgG concentrations [START_REF] Dupin | Asthma and Hypogammaglobulinemia: an Asthma Phenotype with Low Type 2 Inflammation[END_REF][START_REF] Oksenhendler | Infections in 252 patients with common variable immunodeficiency[END_REF][START_REF] Lucas | Infection outcomes in patients with common variable immunodeficiency disorders: relationship to immunoglobulin therapy over 22 years[END_REF]. In asthma patients, serum IgG levels have been found to be significantly lower when asthma exacerbations were associated with positive viral samples [START_REF] Verduyn | Serum IgG Concentrations in Adult Patients Experiencing Virus-Induced Severe Asthma Exacerbations. The journal of allergy and clinical immunology[END_REF].

In literature, it seems that low serum IgG levels tend to be associated with severe asthma and infection-induced asthma exacerbations (mostly viral). We hypothesize that severe eosinophilic asthma patients with the lowest IgG levels may combine both eosinophilic and infection-induced asthma exacerbations. Patients presenting the highest IgG levels might experience mostly eosinophilic exacerbations. This may explain that serum IgG concentrations appear to be a predictive biomarker of response to anti-IL5/IL5Rα therapies in our study. The highest levels of serum IgG found in excellent responders may reflect a "purest" eosinophilic phenotype leading to a more beneficial effect of anti-IL5/IL5Rα therapies. Moreover, poor responders who exhibit the lowest levels of serum IgG may experience a mix between eosinophilic and infectioninduced exacerbations. However, the effect of anti-IL5/IL5Rα therapies on viral infections has not extensively been assessed yet. In a study carried out on mild asthma patients challenged with rhinovirus 16, mepolizumab enhanced local B cells and reduced neutrophils as well as their activation [START_REF] Sabogal Piñeros | Anti-IL-5 in Mild Asthma Alters Rhinovirus-induced Macrophage, B-Cell, and Neutrophil Responses (MATERIAL). A Placebocontrolled, Double-Blind Study[END_REF]. Mepolizumab also enhanced secretory IgA and decreased tryptase in bronchoalveolar lavage [START_REF] Sabogal Piñeros | Anti-IL-5 in Mild Asthma Alters Rhinovirus-induced Macrophage, B-Cell, and Neutrophil Responses (MATERIAL). A Placebocontrolled, Double-Blind Study[END_REF]. Further studies are needed to determine whether anti-IL5/IL5Rα therapies have any effect on virus-induced asthma exacerbations.

The role of serum IgG concentrations as a predictive biomarker of response to anti-IL5/IL5Rα therapies needs further studies. The quality criteria of a reliable biomarker remains unclear in severe asthma. Although, no definition has been established yet, it is obvious that a good predictive biomarker should appear in easy-to-collect fluid and be simply detected or quantified in assays being both affordable and robust [START_REF]Biomarkers on a roll[END_REF][START_REF] Breiteneder | Biomarkers for diagnosis and prediction of therapy responses in allergic diseases and asthma[END_REF]. Regarding the performance of a predictive biomarker, there is no specific definition. It has been established that the area under the ROC curve of blood eosinophil for predicting mepolizumab response ranges from 0.79 to 0.89 [START_REF] Bergantini | Personalized Approach of Severe Eosinophilic Asthma Patients Treated with Mepolizumab and Benralizumab[END_REF][START_REF] Wagener | External validation of blood eosinophils, FE(NO) and serum periostin as surrogates for sputum eosinophils in asthma[END_REF]. In our study, the area under the ROC curve of serum IgG is 0.75. Although the AUC values are close, serum IgG should be confirmed in further prospective clinical studies as a predictive biomarker.

Our study has several limitations. Firstly, the assessment of the response to mepolizumab or benralizumab is based on the GETE score, which is a subjective score with its inherent bias.

However, no objective markers of response have been developed for anti-IL5/IL5Rα therapies.

Furthermore, the GETE score has been validated with omalizumab being the first biological therapy used in severe asthma [START_REF] Bousquet | Predicting and evaluating response to omalizumab in patients with severe allergic asthma[END_REF]. Secondly, whereas the increase in serum IgG concentrations with mepolizumab or benralizumab was statistically significant, it remains uncertain whether this difference is clinically relevant or not. This should be determined in specific clinical trials. However, in order to provide clinically relevant data, we determined if serum IgG concentrations were a useful predictive biomarker of response to anti-IL5/IL5Rα therapies. In fact, we found out that a 9.2 g/l cut-off may be used to determine good and bad responders. Thirdly, Ig quantification is not systematically performed at 6 months in our centre. Thus, the patients' selection based on a quantification at 6 months after the initiation of biological therapies could introduce a bias in the results. Finally, we pooled benralizumab and mepolizumab data but we found no relevant differences between mepolizumab and benralizumab (data not shown).

However, given the small number of patients treated with benralizumab, the comparison between mepolizumab and benralizumab has to be interpreted with caution. Although their mechanisms of action are close but not identical, further comparisons are needed.

In severe asthma patients, mepolizumab or benralizumab are associated with a significant increase in serum IgG concentrations at 6 months. The mechanisms involved in this change over time are not fully understood but could be due to a reduction of OCS use with biological therapies. Thus, further mechanistic studies are required to better elucidate eosinophil-B cells interaction. Moreover, baseline serum IgG concentrations seem to be helpful to predict a good response to anti-IL5/IL5Rα therapies. In order to confirm this hypothesis, this biomarker should be tested in a larger cohort.

[48. 8 -

 8 65.3] years and 25 (50%) patients were female. Thirty-seven (74%) patients were treated with mepolizumab and thirteen (26%) with benralizumab. Mean predictive values of prebronchodilator FEV1 were 71.2 [53.0-88.1].

  .5 [10.0-20.0] mg at baseline vs 6.3 [0.0-10.0] mg at 6 months; p<0.0001). Regarding lung function tests at 6 months, median prebronchodilator FEV1 increased from 71.2 [53.0-88.1]% of mean predicted values to 83.0 [67.0-99.0]% of mean predicted values. At 6 months, 37 (74%) patients were considered as being responders to mepolizumab or benralizumab (Excellent or good response according to GETE score).

Figure 1 :Figure 2 :

 12 Figure 1: Concentrations of serum immunoglobulins at baseline and at 6 months of treatment

3 :Figure 2

 32 Figure 2

Table 1 :

 1 Baseline characteristics of patients hospitalized for asthma exacerbation. ACT: asthma 467 control test; FEV1: forced expiratory volume in 1 second; FVC: forced volume capacity; FeNO: 468 fraction of exhaled nitric oxide; IgE: Immunoglobulin E

Table 2 :

 2 Clinical and functional parameters at baseline and at 6 months of treatment with anti-IL5 therapies. ACT: asthma control test; FEV1: forced expiratory volume in 1 second; FVC: forced volume capacity; FeNO: fraction of exhaled nitric oxide; IgE: Immunoglobulin E

	Baseline	6 months	p

Alain Didier made substantial contributions to conception and design, recruitment of patients and interpretation of data; reviewed the manuscript critically for important intellectual content; gave final approval of the version to be published; and agrees to be accountable for all aspects of the work related to its accuracy or integrity.

Laurent Guilleminault made substantial contributions to conception and design, acquisition of data, and analysis and interpretation of data; drafted the article; gave final approval of the version to be published and agrees to be accountable for all aspects of the work related to its accuracy or integrity.

AstraZeneca, MSD and Novartis, reports grants or fees for consulting from AstraZeneca, GlaxoSmithKline, Novartis and Sanofi-Regeneron, and fees for consulting from Bayer, Chiesi, MSD, not related to the submitted work DB has been an investigator in clinical trials for AstraZeneca, GSK, MSD and Novartis SL, ENS, GP, NG, LR declare no disclosure of interest AD has been an investigator in clinical trials for AstraZeneca, GSK and Novartis, reports grants or fees for consulting from AstraZeneca, GlaxoSmithKline, Novartis and Sanofi-Regeneron, and fees for consulting from Bayer, Chiesi, MSD, outside the submitted work

Authors' contributions:

Simon Lauret made substantial contributions to conception and design, acquisition of data, and analysis and interpretation of data; drafted the article; gave final approval of the version to be published and agrees to be accountable for all aspects of the work related to its accuracy or integrity.

Elise Noel-Savina made substantial contributions to conception and design, recruitment of patients and interpretation of data; reviewed the manuscript critically for important intellectual content; gave final approval of the version to be published; and agrees to be accountable for all aspects of the work related to its accuracy or integrity.

Grégoire Prévot made substantial contributions to conception and design, recruitment of patients and interpretation of data; reviewed the manuscript critically for important intellectual content; gave final approval of the version to be published; and agrees to be accountable for all aspects of the work related to its accuracy or integrity.

Laurent Reber made substantial contributions to conception and design, recruitment of patients and interpretation of data; reviewed the manuscript critically for important intellectual content; gave final approval of the version to be published; and agrees to be accountable for all aspects of the work related to its accuracy or integrity.

Nicolas Guibert made substantial contributions to conception and design, recruitment of patients and interpretation of data; reviewed the manuscript critically for important intellectual content; gave final approval of the version to be published; and agrees to be accountable for all aspects of the work related to its accuracy or integrity.

Danièle Brouquières made substantial contributions to conception and design, recruitment of patients and interpretation of data; reviewed the manuscript critically for important intellectual content; gave final approval of the version to be published; and agrees to be accountable for all aspects of the work related to its accuracy or integrity.


Hugo Lhachemi

Christophe Prieur

Predictor-Based Output Feedback Stabilization of an Input Delayed Parabolic PDE with Boundary Measurement

Keywords: Input delayed reaction-diffusion PDEs, predictor, output feedback, boundary control

This paper is concerned with the output feedback boundary stabilization of general 1-D reaction diffusion PDEs in the presence of an arbitrarily large input delay. We consider the cases of Dirichlet/Neumann/Robin boundary conditions for the both boundary control and boundary condition. The boundary measurement takes the form of a either Dirichlet or Neumann trace. The adopted control strategy is composed of a finite-dimensional observer estimating the first modes of the PDE coupled with a predictor to compensate the input delay. In this context, we show for any arbitrary value of the input delay that the control strategy achieves the exponential stabilization of the closed-loop system, for system trajectories evaluated in H 1 norm (also in L 2 norm in the case of a Dirichlet boundary measurement), provided the dimension of the observer is selected large enough. The reported proof of this result requires to perform both control design and stability analysis using simultaneously the (non-homogeneous) original version of the PDE and one of its equivalent homogeneous representations.

Introduction

Since time delays are ubiquitous in practical applications, feedback control of finite-dimensional systems in the presence of input delays has been extensively studied [START_REF] Artstein | Linear systems with delayed controls: a reduction[END_REF][START_REF]Time-delay systems: an overview of some recent advances and open problems[END_REF]. The extension of this topic to Partial Differential Equations (PDEs) has attracted much attention in the recent years [START_REF] Hashimoto | Stabilization of reaction diffusion equations with state delay using boundary control input[END_REF][START_REF] Nicaise | Stability of the heat and of the wave equations with boundary timevarying delays[END_REF][START_REF] Wang | Delay-dependent exponential stabilization for linear distributed parameter systems with time-varying delay[END_REF]. This paper is concerned with the feedback stabilization of reaction-diffusion PDEs in the presence of an arbitrarily long input delay. One of the very first contributions on this topic was reported in [START_REF] Krstic | Control of an unstable reaction-diffusion PDE with long input delay[END_REF] using a backstepping control design technique (see also [START_REF] Krstic | Compensating actuator and sensor dynamics governed by diffusion PDEs[END_REF] for the related problem of sensor dynamics governed by diffusion PDEs). More recently, the possibility to combine classical spectral reduction methods [START_REF] Coron | Global steadystate controllability of one-dimensional semilinear heat equations[END_REF][START_REF] Coron | Global steadystate stabilization and controllability of 1D semilinear wave equations[END_REF][START_REF] David | Controllability and stabilizability theory for linear partial differential equations: recent progress and open questions[END_REF] (which are based on the fact that the associated eigenfunctions form a Riesz basis) and the design of a classical predictor feedback [START_REF] Artstein | Linear systems with delayed controls: a reduction[END_REF][START_REF] Bekiaris | Compensation of state-dependent input delay for nonlinear systems[END_REF][START_REF] Karafyllis | Predictor feedback for delay systems: Implementations and approximations[END_REF] on a finite-dimensional truncated model of Corresponding author H. Lhachemi. The work of C. Prieur has been partially supported by MIAI@Grenoble Alpes (ANR-19-P3IA-0003)

Email addresses: hugo.lhachemi@centralesupelec.fr (Hugo Lhachemi), christophe.prieur@gipsa-lab.fr (Christophe Prieur).

the original PDE was reported in [START_REF] Prieur | Feedback stabilization of a 1D linear reaction-diffusion equation with delay boundary control[END_REF] in the case of a state-feedback. Extensions of this approach in various directions, also in the context of state-feedback, were reported in [START_REF] Lhachemi | Feedback stabilization of a class of diagonal infinite-dimensional systems with delay boundary control[END_REF][START_REF] Lhachemi | Indomain stabilization of block diagonal infinite-dimensional systems with time-varying input delays[END_REF][START_REF] Lhachemi | PI regulation of a reaction-diffusion equation with delayed boundary control[END_REF]. One of the main advantages of spectral reduction methods for parabolic PDEs is that they allow the design of a finite-dimensional state-feedback, making them particularly relevant for practical applications. However, sole state-feedback control of PDEs is generally inapplicable in practice because the distributed nature of the state makes it essentially impossible to measure. Hence the design of an observer is generally required. Since the plant is a PDE, the observer itself generally takes the form of a PDE synthesized using a backstepping procedure [START_REF] Krstic | Boundary control of PDEs: A course on backstepping designs[END_REF]. Hence the partial statefeedback can be coupled with the observer to ensure the stability of the closed-loop plant; see e.g. [START_REF] Katz | Boundary delayed observer-controller design for reaction-diffusion systems[END_REF] where sufficient LMI conditions are derived with robustness aspects w.r.t. small enough delays. In order to avoid the pitfall of late lumping approximations required for the implementation of observers with infinite dimensional dynamics, a number of works have been devoted to the design of finite-dimensional observer-based control strategies for parabolic PDEs [START_REF] Mark | Finite-dimensional controllers for linear distributed parameter systems: exponential stability using residual mode filters[END_REF][START_REF] Curtain | Finite-dimensional compensator design for parabolic distributed systems with point sensors and boundary input[END_REF][START_REF] Grüne | Finite-dimensional output stabilization of linear diffusion-reaction systems-a small-gain approach[END_REF][START_REF] Harkort | Finitedimensional observer-based control of linear distributed parameter systems using cascaded output observers[END_REF][START_REF] Katz | Constructive method for finite-dimensional observer-based control of 1-D parabolic PDEs[END_REF][START_REF] Lhachemi | Finite-dimensional observer-based PI regulation control of a reaction-diffusion equation[END_REF][START_REF] Lhachemi | Finite-dimensional observer-based boundary stabilization of reaction-diffusion equations with either a Dirichlet or Neumann boundary measurement[END_REF][START_REF] Lhachemi | Local output feedback stabilization of a reaction-diffusion equation with saturated actuation[END_REF][START_REF] Lhachemi | Nonlinear boundary output feedback stabilization of reaction diffusion PDEs[END_REF][START_REF] Sakawa | Feedback stabilization of linear diffusion systems[END_REF][START_REF] Sano | Stability-enhancing control of a coupled transport-diffusion system with Dirichlet actuation and Dirichlet measurement[END_REF]. In this work, we take advantage of the control architecture initially reported in [START_REF] Sakawa | Feedback stabilization of linear diffusion systems[END_REF] augmented with the LMI-based procedure introduced in [START_REF] Katz | Constructive method for finite-dimensional observer-based control of 1-D parabolic PDEs[END_REF]. More precisely, we leverage the enhanced procedure reported in [START_REF] Lhachemi | Finite-dimensional observer-based boundary stabilization of reaction-diffusion equations with either a Dirichlet or Neumann boundary measurement[END_REF] that extends for general reaction-diffusion PDEs the LMI-based approach reported in [START_REF] Katz | Constructive method for finite-dimensional observer-based control of 1-D parabolic PDEs[END_REF] to Dirichlet and/or Neumann boundary control and measurement (see also [START_REF] Katz | Finite-dimensional control of the heat equation: Dirichlet actuation and point measurement[END_REF] with a different approach but limited to Dirichlet measurements).

We address the finite-dimensional observer-based output feedback boundary stabilization of general 1-D reaction diffusion PDEs in the presence of an arbitrarily large input delay. A solution to this control design problem was reported for the first time in [START_REF] Katz | Sub-predictors and classical predictors for finite-dimensional observer-based control of parabolic PDEs[END_REF] by combining a finite-dimensional observer and a predictor (used to compensate the input delay) in the very specific setting of a reaction-diffusion equation with Neumann boundary control, a bounded output operator, and with stability of the closed-loop system assessed in L 2 norm for arbitrarily large value of the input delay. However, the approach developed in [START_REF] Katz | Sub-predictors and classical predictors for finite-dimensional observer-based control of parabolic PDEs[END_REF] and which solely relies on the original (non-homogeneous) representation of the PDE is strongly tailored for the above-mentioned setting and is hardly extendable to other types of boundary conditions (Dirichlet/Robin), to unbounded measurement operators (Dirichlet/Neumann), and to system trajectories evaluated in H 1 norm. This is because, introducing (β n ) n≥1 the coefficients of projection of the boundary control operator into the Hilbert basis formed by the eigenstructures of the underlying Sturm-Liouville operator, the Neumann setting is such that β n = O(1) while the Dirichlet/Robin configurations give in general no better than β n = O( √ λ n ) where λ n are the eigenvalues of the problem that grow in n 2 . This difference of asymptotic behavior has a major impact on the control design procedure since the proof reported in [START_REF] Katz | Sub-predictors and classical predictors for finite-dimensional observer-based control of parabolic PDEs[END_REF], which is limited to the case of a bounded measurement operator with trajectories evaluated in L 2 norm, heavily relies on the convergence of the series β 2 n /λ n that is only granted for Neumann boundary control. This issue becomes even more stringent in the case of an unbounded measurement operator and for system trajectories evaluated in H 1 norm, as it will be further detailed in Remark 3 of this paper after the introduction of the model of the system and its modal decomposition.

In this paper, we completely solve the control design problem of output feedback stabilization of general 1-D reaction-diffusion PDEs in the presence of an arbitrarily large input delay, with Dirichlet/Neumann/Robin boundary control/condition, and with a boundary measurement selected as a either Dirichlet or Neumann trace. The employed control architecture combines a finite-dimensional observer and a predictor. In the case of a Dirichlet (resp. Neumann) measurement, we assess the exponential stability of the closed-loop system in L 2 and H 1 norms (resp. H 1 norm) for arbitrarily large value of the input delay provided the dimension of the observer is selected large enough. This is achieved by leveraging an adequate scaling procedure [START_REF] Lhachemi | Finite-dimensional observer-based boundary stabilization of reaction-diffusion equations with either a Dirichlet or Neumann boundary measurement[END_REF] and by considering simultaneously both the original (non-homogeneous) representation of the PDE and one of its homogeneous versions obtain using a change of variable for control design and Lyapunov stability analysis.

The paper is organized as follows. Notations and properties of Sturm-Liouville operators are presented in Section 2. The problem setting is presented in Section 3 followed by a preliminary spectral reduction of the problem. The case of a Dirichlet measurement is studied in Section 4 while the case of a Neumann measurement is analyzed in Section 5. A numerical example is provided in Section 6. Finally, concluding remarks are formulated in Section 7.

Notation and properties

Notation

Spaces R n are endowed with the Euclidean norm • . The corresponding induced norms of matrices are also denoted by • . For any two vectors X and Y of arbitrary dimensions, col(X, Y ) stands for the vector [X , Y ] . The space of square integrable functions on (0, 1) is denoted by L 2 (0, 1) and is endowed with the usual inner product f, g = 1 0 f (x)g(x) dx and with associated norm denoted by • L 2 . For an integer m ≥ 1, H m (0, 1) denotes the m-order Sobolev space and is equipped with its usual norm • H m . For a symmetric matrix P ∈ R n×n , P 0 (resp. P 0) means that P is positive semi-definite (resp. positive definite).

Properties of Sturm-Liouville operators

Let θ 1 , θ 2 ∈ [0, π/2], p ∈ C 1 ([0, 1]) and q ∈ C 0 ([0, 1]) with p > 0 and q ≥ 0. Let the Sturm-Liouville operator A : D(A) ⊂ L 2 (0, 1) → L 2 (0, 1) be defined by Af = -(pf ) + qf on the domain D(A) = {f ∈ H 2 (0, 1) : c θ1 f (0) -s θ1 f (0) = c θ2 f (1) + s θ2 f (1) = 0} where c θi = cos θ i and s θi = sin θ i . The eigenvalues λ n , n ≥ 1, of A are simple, non negative (due to θ 1 , θ 2 ∈ [0, π/2] and q ≥ 0), and form an increasing sequence with λ n → +∞ as n → +∞. The corresponding unit eigenvectors φ n ∈ L 2 (0, 1) form a Hilbert basis. The domain of the operator A is also characterized by

D(A) = {f ∈ L 2 (0, 1) : n≥1 |λ n | 2 | f, φ n | 2 < +∞}. Let p * , p * , q * ∈ R be such that 0 < p * ≤ p(x) ≤ p * and 0 ≤ q(x) ≤ q * for all x ∈ [0, 1], then it holds 0 ≤ π 2 (n -1) 2 p * ≤ λ n ≤ π 2 n 2 p * + q * for all n ≥ 1 [30]. Moreover if p ∈ C 2 ([0, 1]
), we have (see, e.g., [START_REF] Orlov | On general properties of eigenvalues and eigenfunctions of a Sturm-Liouville operator: comments on "ISS with respect to boundary disturbances for 1-D parabolic PDEs[END_REF]) that φ n (ξ) = O(1) and φ n (ξ) = O( √ λ n ) as n → +∞ for any given ξ ∈ [0, 1]. Assuming further that q > 0, an integration by parts and the continuous embedding

H 1 (0, 1) ⊂ L ∞ (0, 1) show the existence of constants C 1 , C 2 > 0 such that C 1 f 2 H 1 ≤ n≥1 λ n f, φ n 2 = Af, f ≤ C 2 f 2 H 1 (1)
for any f ∈ D(A). The latter inequalities and the Rieszspectral nature of A imply that the series expansion f = n≥1 f, φ n φ n holds in H 2 (0, 1) norm for any f ∈ D(A). Due to the continuous embedding H 1 (0, 1) ⊂ L ∞ (0, 1), we obtain that f (0) = n≥1 f, φ n φ n (0) and f (0) = n≥1 f, φ n φ n (0). We finally define, for any integer

N ≥ 1, R N f = n≥N +1 f, φ n φ n .
3 Problem setting and preliminary spectral reduction

Problem setting

We consider in this paper the input delayed reactiondiffusion system described by

z t (t, x) = (p(x)z x (t, x)) x -q(x)z(t, x) (2a) c θ1 z(t, 0) -s θ1 z x (t, 0) = 0 (2b) c θ2 z(t, 1) + s θ2 z x (t, 1) = u(t -h) (2c) z(0, x) = z 0 (x) (2d)
for t > 0 and x ∈ (0, 1) where

θ 1 , θ 2 ∈ [0, π/2], p ∈ C 2 ([0, 1]
) with p > 0, q ∈ C 0 ([0, 1]), and the input delay h > 0. Here z(t, •) is the state of the PDE at time t, u(t -h) is the delayed version of the command u(t), and z 0 is the initial condition. We assume throughout the paper that u(τ ) = 0 for τ < 0.

Remark 1 Even if we restrict the presentation to parameters θ 1 , θ 2 ∈ [0, π/2], which correspond to the most meaningful configurations from a practical perspective, developments reported in this paper readily extend to the case θ 1 , θ 2 ∈ [0, π) provided q in (5) is selected sufficiently large positive so that (1) still holds, implying in particular λ n ≥ 0 for all n ≥ 1. In this case, one merely needs to modify the change of variable formula ( 6) by the following one: w(t, x) = z(t, x) -

x α c θ 2 +αs θ 2 u(t) where α > 1 is fixed so that c θ2 + αs θ2 = 0.
The system output y(t) ∈ R is selected as the either left Dirichlet trace (in this case

θ 1 ∈ (0, π/2]) y D (t) = z(t, 0) (3)
or left Neumann trace (in this case

θ 1 ∈ [0, π/2)) y N (t) = z x (t, 0). ( 4
)
Without loss of generality, let q ∈ C 0 ([0, 1]) and q c ∈ R be such that q(x) = q(x) -q c , q(x) > 0.

(

) 5
This allows to consider the Sturm-Liouville operator A and its related properties as described in Subsection 2.2.

In particular, since q ≥ 0, the eingenvalues λ n of A are such that λ n ≥ 0. Note however that the actual modes of the reaction-diffusion PDE (2) are given by -λ n + q c , hence a finite number of them may be unstable.

Spectral reduction

In order to obtain an equivalent homogeneous representation of (2), we define the change of variable

w(t, x) = z(t, x) - x 2 c θ2 + 2s θ2 u(t -h). (6)
Introducing v = u, we infer that

u(t) = v(t) (7a) w t (t, x) = (p(x)w x (t, x)) x -q(x)w(t, x) (7b) + a(x)u(t -h) + b(x)v(t -h) (7c) c θ1 w(t, 0) -s θ1 w x (t, 0) = 0 (7d) c θ2 w(t, 1) + s θ2 w x (t, 1) = 0 (7e) w(0, x) = w 0 (x) (7f)
where

a(x) = 1 c θ 2 +2s θ 2 {2p(x)+2xp (x)-x 2 q(x)}, b(x) = -x 2 c θ 2 +2s θ 2
, and

w 0 (x) = z 0 (x) -x 2 c θ 2 +2s θ 2 u(-h) = z 0 (x).
Let the coefficients of projection be defined by

z n (t) = z(t, •), φ n , w n (t) = w(t, •), φ n , a n = a, φ n , and b n = b, φ n .
In particular we have from ( 6) that

w n (t) = z n (t) + b n u(t -h), n ≥ 1. ( 8
)
Using standard arguments, see e.g. [START_REF] Lhachemi | Boundary feedback stabilization of a reaction-diffusion equation with Robin boundary conditions and state-delay[END_REF][START_REF] Mironchenko | Local stabilization of an unstable parabolic equation via saturated controls[END_REF] for details, the projection of ( 7) into the Hilbert basis (φ n ) n≥1 gives

u(t) = v(t) (9a) ẇn (t) = (-λ n + q c )w n (t) + a n u(t -h) + b n v(t -h) (9b)
with w(t, •) = n≥1 w n (t)φ n in L 2 norm for mild solutions and in H 2 norm for classical solutions (see the end of Section 2.2). Using (8) into the latter idendity, the projection of (2) reads

żn (t) = (-λ n + q c )z n (t) + β n u(t -h) (10)
with

β n = a n + (-λ n + q c )b n = p(1){-c θ2 φ n (1) + s θ2 φ n (1)} = O( √ λ n ).
Here we have z(t, •) = n≥1 z n (t)φ n in L 2 norm. Finally, when dealing with classical solutions, the Dirichlet measurement y D (t) given by (3) can be expressed as the series expansion:

y D (t) = z(t, 0) = w(t, 0) = n≥1 w n (t)φ n (0) (11)
while for the Neumann measurement y N (t) given by ( 4) we have:

y N (t) = z x (t, 0) = w x (t, 0) = n≥1 w n (t)φ n (0). ( 12
)
Remark 2 Note that the series expansions [START_REF] Hashimoto | Stabilization of reaction diffusion equations with state delay using boundary control input[END_REF][START_REF] Karafyllis | Predictor feedback for delay systems: Implementations and approximations[END_REF] hold for the coefficients of projection w n , i.e., for the PDE in w coordinates. Such a series expansion does not hold for the coefficients of projection z n , i.e., for the PDE in z coordinates.

Remark 3 The use of a predictor feedback to achieve the boundary stabilization of (2) in the case of Neumann actuation and boundary condition (θ 1 = θ 2 = π/2) was reported first in [START_REF] Katz | Sub-predictors and classical predictors for finite-dimensional observer-based control of parabolic PDEs[END_REF] for a bounded output operator, namely

y(t) = 1 0 c(x)z(t, x) dx with c ∈ L 2 (0, 1
), and for system trajectories evaluated in L 2 norm. In this very specific setting, the authors managed to perform the both control design and stability analysis on the sole representation [START_REF] Harkort | Finitedimensional observer-based control of linear distributed parameter systems using cascaded output observers[END_REF], i.e., for the PDE in original coordinates (2). This approach is strongly tailored for the above-mentioned setting and is hardly extendable to other types of boundary control (Dirichlet/Robin) and to unbounded measurement operators (Dirichlet/Neumann). This is essentially because the consideration of a Neumann actuation yields the most favorable case β n = p(1)φ n (1) = O(1) while any other boundary actuation setting (Dirichlet/Robin) gives in general no better than

β n = O( √ λ n ).
However, one of the crucial points of the L 2 stability analysis performed in [START_REF] Katz | Sub-predictors and classical predictors for finite-dimensional observer-based control of parabolic PDEs[END_REF] relies in the use of the estimate 2

β n z n u ≤ α (β 2 n /λ n )u 2 +α -1 λ n z 2
n , valid for any α > 0, where the convergence of the first series on the RHS holds for the Neumann actuation setting (β n = O(1)) and with a term λ n z 2 n that can be handled, provided its convergence, in the L 2 Lyapunov stability analysis 1 . This approach fails in the case of Dirichlet/Robin actuation settings (

β n = O( √ λ n )).
The situation is even more stringent when trying to assess the stability of the system trajectories in H 1 norm (possibly with unbounded output operators instead of a bounded one) since this would led to a term of the form λ n β n z n that cannot be neither handled with the above approach. In this paper, we completely solve the control design problem for the general reaction diffusion [START_REF] Mark | Finite-dimensional controllers for linear distributed parameter systems: exponential stability using residual mode filters[END_REF] with Dirichlet/Neumann/Robin boundary control/condition and with a measurement selected either as the Dirichlet (3) or Neumann (4) trace. The proposed strategy consists in designing the predictor feedback based on the representation [START_REF] Harkort | Finitedimensional observer-based control of linear distributed parameter systems using cascaded output observers[END_REF] in original coordinates (2) while the stability analysis is performed based on [START_REF] Grüne | Finite-dimensional output stabilization of linear diffusion-reaction systems-a small-gain approach[END_REF] in homogeneous coordinates [START_REF] Curtain | Finite-dimensional compensator design for parabolic distributed systems with point sensors and boundary input[END_REF]. 1 Essentially because λn appearing in

λnz 2 n = λ α n z 2 n
has a power α = 1 that is not larger than the one in the dynamics of the modes [START_REF] Harkort | Finitedimensional observer-based control of linear distributed parameter systems using cascaded output observers[END_REF].

Case of a Dirichlet measurement

We consider in this section the input-delayed reactiondiffusion system (2) for θ 1 ∈ (0, π/2] with Dirichlet measurement (3).

Control strategy

Let δ > 0 be the desired exponential decay rate for the closed-loop system trajectories. Let N 0 ≥ 1 be such that -λ n + q c < -δ < 0 for all n ≥ N 0 + 1. Let N ≥ N 0 +1 be arbitrarily given and that will be specified later. Consider first the following observer dynamics used to estimate the N first modes of the plant in z-coordinates:

ŵn (t) = ẑn (t) + b n u(t -h) (13a) żn (t) = (-λ n + q c )ẑ n (t) + β n u(t -h) (13b) -l n N k=1 ŵk (t)φ k (0) -y D (t) , 1 ≤ n ≤ N 0 żn (t) = (-λ n + q c )ẑ n (t) + β n u(t -h), N 0 + 1 ≤ n ≤ N (13c)
where l n ∈ R are the observer gains.

Remark 4 Dynamics ( 13) constitutes an observer of the N first modes z n of the PDE in (original) z coordinates. However, due to Remark 2 and in view of ( 11), the estimation ŷD (t) of the actual Dirichlet measurement y D (t) is expressed in function of the estimation ŵn of the modes w n of the PDE in homogeneous coordinates w as ŷD (t) = N k=1 ŵk (t)φ k (0). Hence, even if (13) estimates the modes z n in z coordinates, the correction of the error of measurement is done based on the modes w n in w coordinates.

Remark 5

The idea to split the observer dynamics into two parts, one with active correction of the estimation error for the first modes as in (13b) and one without correction of the estimation error as in (13c), roots back to [START_REF] Sakawa | Feedback stabilization of linear diffusion systems[END_REF] in a delay-free context with bounded input and output operators. Since then, such an idea, sometimes referred to as the add of a "Residual Mode Filter" and which was shown to be of paramount importance for ensuring closed-loop stability [START_REF] Mark | Finite-dimensional controllers for linear distributed parameter systems: exponential stability using residual mode filters[END_REF] has been extended in various directions [START_REF] Mark | Finite-dimensional controllers for linear distributed parameter systems: exponential stability using residual mode filters[END_REF][START_REF] Grüne | Finite-dimensional output stabilization of linear diffusion-reaction systems-a small-gain approach[END_REF][START_REF] Harkort | Finitedimensional observer-based control of linear distributed parameter systems using cascaded output observers[END_REF][START_REF] Katz | Constructive method for finite-dimensional observer-based control of 1-D parabolic PDEs[END_REF][START_REF] Lhachemi | Finite-dimensional observer-based PI regulation control of a reaction-diffusion equation[END_REF][START_REF] Lhachemi | Finite-dimensional observer-based boundary stabilization of reaction-diffusion equations with either a Dirichlet or Neumann boundary measurement[END_REF][START_REF] Lhachemi | Local output feedback stabilization of a reaction-diffusion equation with saturated actuation[END_REF][START_REF] Lhachemi | Nonlinear boundary output feedback stabilization of reaction diffusion PDEs[END_REF][START_REF] Sakawa | Feedback stabilization of linear diffusion systems[END_REF][START_REF] Sano | Stability-enhancing control of a coupled transport-diffusion system with Dirichlet actuation and Dirichlet measurement[END_REF].

Due to the input delay h > 0, we need to introduce a predictor component. To do so, let ẐN0 = ẑ1 . . . ẑN0 , A 0 = diag(-λ 1 + q c , . . . , -λ N0 + q c ), and B 0 = β 1 . . . β N0 . We can now introduce the following Artstein transformation [START_REF] Artstein | Linear systems with delayed controls: a reduction[END_REF]:

ẐN0 A (t) = e A0h ẐN0 (t) + t t-h e A0(t-τ ) B 0 u(τ ) dτ. (14)
Then the control is defined as the predictor feedback [START_REF] Karafyllis | Predictor feedback for delay systems: Implementations and approximations[END_REF]:

u(t) = K ẐN0 A (t), t ≥ 0 ( 15
)
where K ∈ R 1×N0 is the feedback gain.

Remark 6

The well-posedness of the closed-loop system composed of the plant [START_REF] Curtain | Finite-dimensional compensator design for parabolic distributed systems with point sensors and boundary input[END_REF] in homogeneous coordinates w, the Dirichlet measurement (3), and the controller [START_REF] Katz | Constructive method for finite-dimensional observer-based control of 1-D parabolic PDEs[END_REF][START_REF] Katz | Finite-dimensional control of the heat equation: Dirichlet actuation and point measurement[END_REF][START_REF] Katz | Sub-predictors and classical predictors for finite-dimensional observer-based control of parabolic PDEs[END_REF] in terms of classical solutions for initial condition z 0 ∈ H 2 (0, 1) so that c θ1 z 0 (0) -s θ1 z 0 (0) = 0 and c θ2 z 0 (1) + s θ2 z 0 (1) = 0, with null control in negative times (u(τ ) = 0 for τ < 0) and zero initial condition for the observer (ẑ n (0) = 0), is a direct consequence of [31, Thm. 6.3.1 and 6.3.3] and the invertibility of the Artstein transformation [START_REF] Bresch-Pietri | New formulation of predictors for finite-dimensional linear control systems with input delay[END_REF] using a classical induction argument. Having obtained classical solutions based on the homogeneous representation [START_REF] Curtain | Finite-dimensional compensator design for parabolic distributed systems with point sensors and boundary input[END_REF], standard arguments [8, Sec. 3.3] using the change of variable formula [START_REF] Coron | Global steadystate stabilization and controllability of 1D semilinear wave equations[END_REF] give the existence of classical solutions for the closed-loop system with the plant (2) expressed in original z coordinates.

In preparation of the statement of the main result, we define the matrices 15) and the observer gain L whose coefficients l n appear in [START_REF] Katz | Constructive method for finite-dimensional observer-based control of 1-D parabolic PDEs[END_REF] are computed so that A 0 + B 0 K and A 0 -LC 0 are Hurwitz with eigenvalues that have a real part strictly less than -δ < 0. To complete the control design procedure, it merely remains to select adequately the dimension N of the observer to ensure the stability of the closed-loop system with exponential decay rate δ > 0.

A 1 = diag(-λ N0+1 + q c , . . . , -λ N + q c ), B1 = β N0+1 /λ N0+1 . . . β N /λ N , C 0 = φ 1 (0) . . . φ N0 (0) , C1 = φ N 0 +1 (0) √ λ N 0 +1 . . . φ N (0) √ λ N , L = l 1 . . . l N0 , F =        A 0 + B 0 K e A0h LC 0 0 e A0h L C1 0 A 0 -LC 0 0 -L C1 B1 K 0 A 1 0 0 0 0 A 1        , L =        e A0h L -L 0 0        E = A 0 + B 0 K e A0h LC

Main stability results

Theorem 8 Let θ 1 ∈ (0, π/2], θ 2 ∈ [0, π/2], p ∈ C 2 ([0, 1]) with p > 0, and q ∈ C 0 ([0, 1]). Let q ∈ C 0 ([0, 1]) and q c ∈ R be such that (5) holds. Let δ > 0 and N 0 ≥ 1 be such that -λ n + q c < -δ for all n ≥ N 0 + 1. Let K ∈ R 1×N0 and L ∈ R N0 be such that A 0 + B 0 K and A 0 -LC 0 are Hurwitz with eigenvalues that have a real part strictly less than -δ < 0. Let h > 0 be given. For a given N ≥ N 0 + 1, assume that there exist P 0, Q 1 , Q 2 0, α > 1, and β, γ > 0 such that

Θ 1 0, Θ 2 ≤ 0, R 1 0, R 2 0 ( 16
)
where

Θ 1 = F P + P F + 2δP + Q1 P L L P -β + E Q 2 E (17a) Θ 2 = 2γ -1 - 1 α λ N +1 + q c + δ + βM φ (17b) R 1 = -e -2δh Q 1 + αγ R N a 2 L 2 K K (17c) R 2 = -e -2δh Q 2 + αγ R N b 2 L 2 K K (17d) with Q1 = diag(Q 1 , 0, 0, 0) and M φ = n≥N +1 |φn(0)| 2 λn < +∞.
Then there exists a constant M > 0 such that for any initial condition z 0 ∈ H 2 (0, 1) so that c θ1 z 0 (0) -s θ1 z 0 (0) = 0 and c θ2 z 0 (1) + s θ2 z 0 (1) = 0, the trajectories of the closed-loop system composed of the plant (2), the Dirichlet measurement (3), and the controller (13-15) with null control in negative times (u(τ ) = 0 for τ < 0) and zero initial condition for the observer (ẑ n (0) = 0) satisfy z(t, •) 2

H 1 + sup τ ∈[t-h,t] |u(τ )| 2 + N n=1 ẑn (t) 2 ≤ M e -2δt z 0 2
H 1 for all t ≥ 0. Moreover, for any given h > 0, the constraints ( 16) are always feasible for N selected large enough.

Proof. We first write a finite dimensional model capturing the N first modes of the PDE and the dynamics (13-15) of the output feedback controller. We define e n = z n -ẑn for all 1 ≤ n ≤ N . In view of (13a-13b) and based on ( 8) and [START_REF] Hashimoto | Stabilization of reaction diffusion equations with state delay using boundary control input[END_REF], we obtain that żn = (-

λ n + q c )ẑ n + β n u(• -h) + l n N k=1 φ k (0)e k + l n ζ (18) for 1 ≤ n ≤ N 0 where ζ(t) = n≥N +1 w n (t)φ n (0). Introducing E N0 = e 1 .
. . e N0 , the scaled error ẽn = √

λ n e n (see [START_REF] Lhachemi | Finite-dimensional observer-based boundary stabilization of reaction-diffusion equations with either a Dirichlet or Neumann boundary measurement[END_REF]), and ẼN-N0 = ẽN0+1 . . . ẽN , we deduce that

ŻN0 = A 0 ẐN0 +B 0 u(•-h)+LC 0 E N0 +L C1 ẼN-N0 +Lζ. ( 19
) Taking first the time derivative of ( 14) and then inserting [START_REF] Katz | Sub-predictors and classical predictors for finite-dimensional observer-based control of parabolic PDEs[END_REF] and ( 19), we infer that

ŻN0 A = (A 0 + B 0 K) ẐN0 A (20) + e A0h LC 0 E N0 + L C1 ẼN-N0 + Lζ .
Consider now (13c). Defining the scaled estimation zn = ẑn /λ n and ZN-N0 = zN0+1 . . . zN , this latter dynamics can be written as

ŻN-N0 = A 1 ZN-N0 + B1 u(t -h).
In order to eliminate the input delay, we consider the second following Artstein transformation:

ZN-N0 A (t) = e A1h ZN-N0 (t) + t t-h e A1(t-τ ) B1 u(τ ) dτ (21)
which implies, along with [START_REF] Katz | Sub-predictors and classical predictors for finite-dimensional observer-based control of parabolic PDEs[END_REF], that

ŻN-N0

A = A 1 ZN-N0 A + B1 K ẐN0 A . (22)
Combining ( 10) and (13b-13c), the error dynamics reads

ĖN0 = (A 0 -LC 0 )E N0 -L C1 ẼN-N0 -Lζ, (23a) ĖN-N0 = A 1 ẼN-N0 . (23b)
Therefore, defining the vector

X = col ẐN0 A , E N0 , ZN-N0 A , ẼN-N0 , (24)
we infer from ( 20) and [START_REF] Lhachemi | Local output feedback stabilization of a reaction-diffusion equation with saturated actuation[END_REF][START_REF] Lhachemi | Nonlinear boundary output feedback stabilization of reaction diffusion PDEs[END_REF] that

Ẋ = F X + Lζ. (25)
Finally, defining X = col (X, ζ) and based on ( 15) and ( 20), we also have

u = KX, v = u = K ŻN0 A , ŻN0 A = E X. ( 26
)
We can now perform the stability analysis. Consider the functional defined by V (t) = V 0 (t)+V 1 (t)+V 2 (t) where

V 0 (t) = X(t) P X(t) + γ n≥N +1 λ n w n (t) 2 (27a) V 1 (t) = t (t-h)+ e -2δ(t-s) ẐN0 A (τ ) Q 1 ẐN0 A (τ )dτ (27b) V 2 (t) = t (t-h)+ e -2δ(t-s) ŻN0 A (τ ) Q 2 ŻN0 A (τ ) dτ (27c)
with (t -h) + = max(t -h, 0). The computation of the time derivative of V for t > h gives V ≤ X F P + P F P L L P 0

X + 2γ n≥N +1 λ n (-λ n + q c )w 2 n + 2γ n≥N +1 λ n {a n u(• -h) + b n v(• -h)}w n -2δV 1 -2δV 2 + ( ẐN0 A ) Q 1 ẐN0 A -e -2δh ẐN0 A (• -h) Q 1 ẐN0 A (• -h) + ( ŻN0 A ) Q 2 ŻN0 A -e -2δh ŻN0 A (• -h) Q 2 ŻN0 A (• -h).
Using Young inequality and invoking [START_REF] Katz | Sub-predictors and classical predictors for finite-dimensional observer-based control of parabolic PDEs[END_REF], we obtain that

2 n≥N +1 λ n a n u(• -h)w n ≤ 1 α n≥N +1 λ 2 n w 2 n + α R N a 2 L 2 u(• -h) 2 ≤ 1 α n≥N +1 λ 2 n w 2 n + α R N a 2 L 2 ẐN0 A (• -h) K K ẐN0 A (• -h)
for any α > 0 and, similarly,

2 n≥N +1 λ n b n v(• -h)w n ≤ 1 α n≥N +1 λ 2 n w 2 n + α R N b 2 L 2 ŻN0 A (• -h) K K ŻN0 A (• -h)
Combining the latter estimates and using [START_REF] Lhachemi | PI regulation of a reaction-diffusion equation with delayed boundary control[END_REF], we obtain

V + 2δV ≤ X F P + P F + 2δP + Q1 P L L P 0 + E Q 2 E X + 2γ n≥N +1 λ n -1 - 1 α λ n + q c + δ w 2 n + ẐN0 A (• -h) R 1 ẐN0 A (• -h) + ŻN0 A (• -h) R 2 ŻN0 A (• -h). Recalling that ζ = n≥N +1 w n φ n (0) we have ζ 2 ≤ M φ n≥N +1 λ n w 2
n . Hence, we obtain, for any β > 0,

V + 2δV ≤ X Θ 1 X + n≥N +1 λ n Γ n w 2 n + ẐN0 A (• -h) R 1 ẐN0 A (• -h) + ŻN0 A (• -h) R 2 ŻN0 A (• -h) ( 28
)
where Γ n = 2γ -1 -1 α λ n + q c + δ +βM φ . For α > 1, we have Γ n ≤ Γ N +1 = Θ 2 for all n ≥ N + 1. Thus we infer from ( 16) that V + 2δV ≤ 0 for all t > h, implying that V (t) ≤ e -2δ(t-h) V (h) for all t ≥ h.

Computing now the time derivative of V for t ∈ (0, h) for which u(• -h) and v(• -h) are zero, and proceeding similarly to the previous paragraph, we infer that

V + 2δV ≤ X Θ 1 X + n≥N +1 λ n Γ n w 2 n with Γ n = 2γ {-λ n + q c + δ} + βM φ ≤ Γ n ≤ Θ 2 for n ≥ N + 1.
Hence we obtain that V + 2δV ≤ 0 on (0, h) thus V (t) ≤ e -2δt V (0) for all 0 ≤ t ≤ h. Combining this estimate with the result of the previous paragraph, we infer that V (t) ≤ e -2δt V (0) for all t ≥ 0. The claimed stability estimate now easily follows from the definition of V , the estimates (1), the control law [START_REF] Katz | Sub-predictors and classical predictors for finite-dimensional observer-based control of parabolic PDEs[END_REF], and the two Artstein transformations ( 14) and [START_REF] Lhachemi | Finite-dimensional observer-based PI regulation control of a reaction-diffusion equation[END_REF].

It remains to show that the constraints Θ 1 , Θ 2 , R 1 , R 2 0 are feasible when selecting N ≥ N 0 + 1 large enough. Regarding the matrix F + δI, we note that (i) A 0 + B 0 K + δI and A 0 -LC 0 + δI are Hurwitz; (ii) e (A1+δI)t ≤ e -κ0t for all t ≥ 0 with κ 0 = λ N0+1 -q c -δ > 0 is independent of N ; and (iii) e A0h L C1 ≤ e A0h L C1 , L C1 ≤ L C1 and B1 K ≤ B1 K where e A0h , K, and L are independent of the number of observed modes N while C1 = O(1) and B1 = O(1) when N → +∞. Hence, applying the Lemma reported in Appendix to the matrix F + δI, we infer that the solution P 0 to F P + P F + 2δP = -I is such that P = O(1) as N → +∞. Note also that L is a constant independent of N while M φ = O( 1) and E = O(1) as N → +∞. We fix arbitrarily the value of α > 1 and we set

β = √ N , γ = 1/N , Q 1 = 2e 2δh αγ R N a 2 L 2 K K, and Q 2 = 2e 2δh αγ R N b 2 L 2 K K.
Hence, using in particular Schur complement, we infer that (16) hold for [START_REF] Lhachemi | Boundary feedback stabilization of a reaction-diffusion equation with Robin boundary conditions and state-delay[END_REF]. The term V 0 presents a first term that accounts for the dynamics of the observer as well as the dynamics of the N first modes of the (original) PDE in z coordinates. However, the second term, which accounts for the residual modes n ≥ N + 1, is expressed in homogeneous w coordinates. This point is key in the success of the stability assessment of the previous theorem. The term V 1 is introduced as in [START_REF] Katz | Sub-predictors and classical predictors for finite-dimensional observer-based control of parabolic PDEs[END_REF] in order to compensate the term u(• -h) appearing in the time derivative of w n , see [START_REF] Grüne | Finite-dimensional output stabilization of linear diffusion-reaction systems-a small-gain approach[END_REF]. Moreover, since the modes n ≥ N + 1 are captured by w n in w coordinates, the time derivative of w n also implies the occurrence of v = u = K ŻN0

N ≥ N 0 + 1 selected large enough. 2 Remark 9 Recall the definition of V (t) = V 0 (t) + V 1 (t) + V 2 (t) with V i defined by
A , see again [START_REF] Grüne | Finite-dimensional output stabilization of linear diffusion-reaction systems-a small-gain approach[END_REF]. This latter term is handled in the stability analysis by the introduction of the term V 2 .

While Theorem 8 assesses the stability of the closed-loop in H 1 norm, the following theorem states a similar result but for trajectories evaluated in L 2 norm.

Theorem 10 Let θ 1 ∈ (0, π/2], θ 2 ∈ [0, π/2], p ∈ C 2 ([0, 1]
) with p > 0, and q ∈ C 0 ([0, 1]). Let q ∈ C 0 ([0, 1]) and q c ∈ R be such that (5) holds. Let δ > 0 and N 0 ≥ 1 be such that -λ n + q c < -δ for all n ≥ N 0 + 1. Let K ∈ R 1×N0 and L ∈ R N0 be such that A 0 + B 0 K and A 0 -LC 0 are Hurwitz with eigenvalues that have a real part strictly less than -δ < 0. Let h > 0 be given. For a given N ≥ N 0 + 1, assume that there exist P 0, Q 1 , Q 2 0, and α, β, γ > 0 such that

Θ 1 0, Θ 2 ≤ 0, Θ 3 ≥ 0, R 1 0, R 2 0 (29)
where Θ 1 , R 1 , R 2 are defined by (17a), (17c), and (17d), respectively, while

Θ 2 = 2γ -λ N +1 + q c + δ + 1 α + βM φ λ 3/4 N +1 Θ 3 = 2γ - βM φ λ 1/4 N +1 with Q1 = diag(Q 1 , 0, 0, 0) and M φ = n≥N +1 |φn(0)| 2 λ 3/4 n < +∞.
Then there exists a constant M > 0 such that for any initial condition z 0 ∈ H 2 (0, 1) so that c θ1 z 0 (0) -s θ1 z 0 (0) = 0 and c θ2 z 0 (1) + s θ2 z 0 (1) = 0, the trajectories of the closed-loop system composed of the plant (2), the Dirichlet measurement (3), and the controller (13-15) with null control in negative times (u(τ ) = 0 for τ < 0) and zero initial condition for the observer (ẑ n (0) = 0) satisfy z(t,

•) 2 L 2 + sup τ ∈[t-h,t] |u(τ )| 2 + N n=1 ẑn (t) 2 ≤ M e -2δt z 0 2
L 2 for all t ≥ 0. Moreover, for any given h > 0, the constraints (29) are always feasible for N selected large enough.

Proof. Consider the functional defined by

V (t) = V 0 (t) + V 1 (t) + V 2 (t) where V 0 (t) = X(t) P X(t) + γ n≥N +1 w 2 n while V 1 , V 2
are defined as in [START_REF] Lhachemi | Boundary feedback stabilization of a reaction-diffusion equation with Robin boundary conditions and state-delay[END_REF]. Proceeding as in the proof of Theorem 8 but replacing the estimate of ζ by the following:

ζ 2 ≤ M φ n≥N +1 λ 3/4 n w 2 n , we infer that V + 2δV ≤ X Θ 1 X + n≥N +1 Γ n w 2 n + ẐN0 A (• -h) R 1 ẐN0 A (• -h) + ŻN0 A (• -h) R 2 ŻN0 A (• -h) holds for t > h with Γ n = 2γ -λ n + q c + δ + 1 α + βM φ λ 3/4 n . For n ≥ N + 1 we note that λ 3/4 n = λ n /λ 1/4 n ≤ λ n /λ 1/4 N +1 hence Γ n ≤ -Θ 3 λ n + 2γ q c + δ + 1 α ≤ -Θ 3 λ N +1 + 2γ q c + δ + 1 α = Θ 2 ≤ 0
where we used that Θ 3 ≥ 0. Therefore, the assumptions imply that V + 2δV ≤ 0 for t > h. Similarly to the proof of Theorem 8, it can also be seen that V + 2δV ≤ 0 for t ∈ (0, h). Gathering these two results together, we infer that V (t) ≤ e -2δt V (0) for all t ≥ 0, implying the claimed stability estimate.

Regarding the feasibility of the constraints [START_REF] Nicaise | Stability of the heat and of the wave equations with boundary timevarying delays[END_REF] for N ≥ N 0 + 1 large enough, this can be achieved following the same procedure as in the proof of Theorem 8 with α > 0 arbitrarily fixed, β = N 1/8 , and γ = 1/N 1/4 . 2

Remark 11 Let N ≥ N 0 +1 be a given number of modes to be observed. When fixing the value of α > 1 (resp. α > 0), the constraints ( 16) from Theorem 8 (resp. the constraints constraints (29) from Theorem 10) take the form of LMIs for which efficient solvers exist. Moreover, as shown in the proof of the two theorems, the resulting LMI constraints remain feasible (when fixing arbitrarily the value of α) for N selected large enough.

Case of a Neumann measurement

We consider in this section the input-delayed reactiondiffusion system (2) for θ 1 ∈ [0, π/2) with Neumann measurement (4).

Control strategy

Let δ > 0 and N 0 ≥ 1 be such that -λ n + q c < -δ < 0 for all n ≥ N 0 + 1. Let N ≥ N 0 + 1 be arbitrarily given. Consider first the following observer dynamics used to estimate the N first modes of the plant in z-coordinates:

ŵn (t) = ẑn (t) + b n u(t -h) (30a) żn (t) = (-λ n + q c )ẑ n (t) + β n u(t -h) (30b) -l n N k=1 ŵk (t)φ k (0) -y N (t) , 1 ≤ n ≤ N 0 żn (t) = (-λ n + q c )ẑ n (t) + β n u(t -h), N 0 + 1 ≤ n ≤ N (30c)
where l n ∈ R are the observer gains. With the same notations that the ones of the previous section and introducing the Artstein transformation ( 14), the control input is defined as

u(t) = K ẐN0 A (t), t ≥ 0 ( 31
)
where K ∈ R 1×N0 is the feedback gain.

Remark 12

The well-posedness of the closed-loop system in terms of classical solutions for initial condition z 0 ∈ H 2 (0, 1) so that c θ1 z 0 (0) -s θ1 z 0 (0) = 0 and c θ2 z 0 (1) + s θ2 z 0 (1) = 0, with null control in negative times (u(τ ) = 0 for τ < 0) and zero initial condition for the observer (ẑ n (0) = 0), follows the same arguments as Remark 6.

We finally introduce the same matrices as at the end of Subsection 4.1 except that we replace the definitions of C 0 , C1 by C 0 = φ 1 (0) . . . φ N0 (0) and C1 = φ N0+1 (0)/λ N0+1 . . . φ N (0)/λ N .

Main stability result

Theorem 13 Let θ 1 ∈ [0, π/2), θ 2 ∈ [0, π/2], p ∈ C 2 ([0, 1]
) with p > 0, and q ∈ C 0 ([0, 1]). Let q ∈ C 0 ([0, 1]) and q c ∈ R be such that (5) holds. Let δ > 0 and N 0 ≥ 1 be such that -λ n + q c < -δ for all n ≥ N 0 + 1. Let K ∈ R 1×N0 and L ∈ R N0 be such that A 0 + B 0 K and A 0 -LC 0 are Hurwitz with eigenvalues that have a real part strictly less than -δ < 0. Let h > 0 be given. For a given N ≥ N 0 + 1, assume that there exist P 0, Q 1 , Q 2 0, ∈ (0, 1/2], α > 1, and β, γ > 0 such that

Θ 1 0, Θ 2 ≤ 0, Θ 3 ≥ 0, R 1 0, R 2 0 (32)
where Θ 1 , R 1 , R 2 are defined by (17a), (17c), and (17d), respectively, while

Θ 2 = 2γ -1 - 1 α λ N +1 + q c + δ + βM φ ( )λ 1/2+ N +1 Θ 3 = 2γ 1 - 1 α - βM φ ( ) λ 1/2- N +1 with Q1 = diag(Q 1 , 0, 0, 0) and M φ ( ) = n≥N +1 |φ n (0)| 2 λ 3/2+ n < +∞.
Then there exists a constant M > 0 such that for any initial condition z 0 ∈ H 2 (0, 1) so that c θ1 z 0 (0) -s θ1 z 0 (0) = 0 and c θ2 z 0 (1) + s θ2 z 0 (1) = 0, the trajectories of the closed-loop system composed of the plant (2), the Neumann measurement (4), and the controller [START_REF] Orlov | On general properties of eigenvalues and eigenfunctions of a Sturm-Liouville operator: comments on "ISS with respect to boundary disturbances for 1-D parabolic PDEs[END_REF][START_REF] Pazy | Semigroups of linear operators and applications to partial differential equations[END_REF] with null control in negative times (u(τ ) = 0 for τ < 0) and zero initial condition for the observer (ẑ n (0

) = 0) satisfy z(t, •) 2 H 1 + sup τ ∈[t-h,t] |u(τ )| 2 + N n=1 ẑn (t) 2 ≤ M e -2δt z 0 2
H 1 for all t ≥ 0. Moreover, for any given h > 0, the constraints (32) are always feasible for N selected large enough.

Proof. Proceeding as in the first part of the proof of Theorem 8 but with ẽn = λ n e n and ζ(t) = n≥N +1 w n (t)φ n (0), we infer that (25) holds.

Consider the functional defined by V (t) = V 0 (t)+V 1 (t)+ V 2 (t) where V 0 , V 1 , V 2 are defined by [START_REF] Lhachemi | Boundary feedback stabilization of a reaction-diffusion equation with Robin boundary conditions and state-delay[END_REF]. Proceeding as in the proof of Theorem 8 but replacing the estimate of ζ by:

ζ 2 ≤ M φ ( ) n≥N +1 λ 3/2+ n w 2 n , we infer that V + 2δV ≤ X Θ 1 X + n≥N +1 λ n Γ n w 2 n + ẐN0 A (• -h) R 1 ẐN0 A (• -h) + ŻN0 A (• -h) R 2 ŻN0 A (• -h) holds for t > h with Γ n = 2γ -(1 -1 α )λ n + q c + δ + βM φ ( )λ 1/2+ n . For n ≥ N + 1 we note that λ 1/2+ n = λ n /λ 1/2- n ≤ λ n /λ 1/2- N +1 hence Γ n ≤ -Θ 3 λ n + 2γ {q c + δ} ≤ -Θ 3 λ N +1 + 2γ {q c + δ} = Θ 2 ≤ 0
where we used that Θ 3 ≥ 0. Therefore, the assumptions imply that V + 2δV ≤ 0 for t > h. Similarly to the proof of Theorem 8, it can also be seen that V + 2δV ≤ 0 for t ∈ (0, h). Gathering these two results together, we infer that V (t) ≤ e -2δt V (0) for all t ≥ 0, implying the claimed stability estimate.

Regarding the feasibility of the constraints (32) for N ≥ N 0 + 1 large enough, this is achieved following the same procedure as in the proof of Theorem 8 with α > 1 arbitrarily fixed, = 1/8, β = N 1/8 , and γ = 1/N

hence ζ 2 ≤ M φ ( ) n≥N +1 λ 3/2+ n w 2 n with M φ ( ) = n≥N +1 |φ n (0)| 2 λ 3/2+ n
where all the terms are finite provided ∈ (0, 1/2]. However a term in λ 3/2+ n cannot be asymptotically dominated by a term in λ n . Hence the procedure of Theorem 10 for a Dirichlet measurement cannot be used anymore in the case of a Neumann measurement.

Numerical illustration

We consider the parameters p = 1, q = -5, θ 1 = π/5, θ 2 = 0 (Dirichlet boundary control), and the input delay h = 1 s. In this setting, the reaction-diffusion PDE described by ( 2) is open-loop unstable. We set the feedback gain K = -0.6950. The observer gain is set as L = 1.7695 in the case of the Dirichlet measurement (3) while L = 1.2856 in the case of the Neumann measurement [START_REF] Bresch-Pietri | New formulation of predictors for finite-dimensional linear control systems with input delay[END_REF]. With δ = 0.5 and for the Dirichlet measurement (3), the constraints of Theorems 8 and 10 are feasible for N = 2 modes estimated by the observer, ensuring the exponential stability of the closed-loop system in H 1 and L 2 norms. Considering now the case of the Neumann boundary measurement (4), the application of the constraints of Theorem 13 are found feasible for N = 6 modes estimated by the observer, ensuring the exponential stability of the closed-loop system in H 1 norm.

For numerical illustration, we consider the Dirichlet measurement (3) along with the initial condition z 0 (x) = 10x 2 (x -1). The evolution of the closed-loop system is depicted in Fig. 1. We observe the exponential decay of the both state of the PDE and observation error in spite of the h = 1 s input delay. This is compliant with the theoretical prediction of Theorems 8 and 10.

Conclusion

This paper solved the boundary stabilization problem of general 1-D reaction-diffusion PDEs in the presence of an arbitrarily large input delay. The approach is very general as it covers the cases of Dirichlet/Neumann/Robin boundary control/condition with Dirichlet/Neumann boundary measurement and for system trajectories evaluated in H 1 norm (also in L 2 norm for Dirichlet measurement). The control strategy couples of finite-dimensional observer that observes a finite number of modes of the PDE and a predictor to compensate the arbitrarily long input delay. Future research directions may be concerned with nonlinear PDEs and non collocated boundary conditions.

A Useful lemma

The following lemma is borrowed from [START_REF] Lhachemi | Finite-dimensional observer-based boundary stabilization of reaction-diffusion equations with either a Dirichlet or Neumann boundary measurement[END_REF]Appendix] and is a generalization of a result presented in [START_REF] Katz | Constructive method for finite-dimensional observer-based control of 1-D parabolic PDEs[END_REF]. We assume that there exist constants C 0 , κ 0 > 0 such that e M N 33 t ≤ C 0 e -κ0t and e M N 44 t ≤ C 0 e -κ0t for all t ≥ 0 and all N ≥ 1. Moreover, we assume that there exists a constant C 1 > 0 such that M N 14 ≤ C 1 , M N 24 ≤ C 1 , and M N 31 ≤ C 1 for all N ≥ 1. Then there exists a constant C 2 > 0 such that, for any N ≥ 1, there exists a symmetric matrix P N ∈ R n+m+2N with P N 0 such that (F N ) P N + P N F N = -I and P N ≤ C 2 .

Remark 7

 7 0 0 e A0h L C1 e A0h L , and K = K 0 0 0 . Since A 0 is diagonal, the Hautus test combined with the boundary conditions involved in the definition of D(A) show that the pairs (A 0 , B 0 ) and (A 0 , C 0 ) both satisfy the Kalman condition. The same remark applies in the case of the Neumann boundary measurement (4) studied in Section 5. Subsequently, the feedback gain K from (

Fig. 1 .

 1 Fig. 1. Time evolution of the closed-loop system for Dirichlet measurement yD(t) = z(t, 0) with delay h = 1 s

Lemma 16

 16 Let n, m, N ≥ 1, M 11 ∈ R n×n and M 22 ∈ R m×m Hurwitz, M 12 ∈ R n×m , M N 14 ∈ R n×N , M N 24 ∈ R m×N , M N 31 ∈ R N ×n , M N 33 , M N 44 ∈ R N ×N , and

  Remark 15 In the case of a Dirichlet measurement, it was possible to propose a L 2 version of the stability result, namely Theorem 10. However, the approach used in the proof of this latter result fails when trying to study the trajectories in L 2 norm for a Neumann measurement. This is because, in this setting, ζ = n≥N +1 w n φ n (0)

3/16

. 2 Remark 14 Similarly to Remark 11, LMIs can be derived from the constraints (13) of Theorem 13 by fixing the values of α > 1 and ∈ (0, 1/2]. Moreover, as shown in the proof of the theorem, the subsequent LMI conditions with any fixed α > 1 and when setting = 1/8 remain feasible for a number of observed modes N selected large enough.