Hélène Vallet

Pascal Seve

Lucie Biard

Jean Baptiste Fraison

Philip Bielefeld

Laurent Perard

Boris Bienvenu

Sébastien Abad

Aude Rigolet

Alban Deroux

Damien Sene

Antoinette Perlat

Isabelle Marie

Elodie Feurer

Eric Hachulla

Olivier Fain

Gaëlle Clavel

Sophie Riviere

Pierre-Alban Bouche

Julie Gueudry

Gregory Pugnet

Phuc L E Hoang

Matthieu Resche Rigon

Patrice Cacoub

Bahram Bodaghi

Dr David Saadoun
email: david.saadoun@aphp.fr

INFLIXIMAB VERSUS ADALIMUMAB IN THE TREATMENT OF REFRACTORY INFLAMMATORY UVEITIS: MULTICENTER STUDY FROM THE FRENCH UVEITIS NETWORK

Keywords: anti-TNFα, refractory inflammatory uveitis, Efficacy, Safety Words count:

Objectives:

To analyze the factors associated with response to anti-TNFα and to compare the efficacy and safety of infliximab (IFX) and adalimumab (ADA) in patients with refractory non infectious uveitis.

Methods: Observational multicenter study including 160 patients [median age of 31 years with 39% of men] with refractory uveitis treated with anti-TNFα [IFX 5mg/kg at weeks 0, 2, 6 and every 5-6 weeks (n=98) and ADA 40mg/14days (n=62)]. Factors associated to complete response were assessed in multivariate analysis. Comparison between efficacy and safety of IFX and ADA was performed using a propensity score approach accounting for baseline characteristics.

Results: Main etiologies of uveitis included Behçet's disease (36%), juvenile idiopathic arthritis (22%), spondylarthropathies (10%) and sarcoidosis (6%). The overall response rate at 6 and 12 months was of 87% (26% of complete response) and 93% (28% of complete response), respectively. The median time to complete response was 2 (0-12) months. In multivariate analysis, Behçet's disease .71], p=0.004) and a number of uveitis flares before anti-TNFα greater than 5 (HR=1.97 [1.02-3.84], p=0.045) were associated with complete response to anti-TNFα. Side effects were reported in 28% of patients, including 12% of serious adverse events. IFX and ADA did not differ significantly in terms of occurrence of complete response (SHR=0.65 [0.25;1.71], p=0.39), serious side effects (SHR= 0.22 [0.04-1.25], p=0.089) or event free survival (SHR=0.55 [0.28;1.08], p=0.083).

Conclusions: Anti-TNFα are highly effective in refractory inflammatory uveitis. Behçet's disease is associated with increase odds of response. IFX and ADA seem equivalent in terms of efficacy.

INTRODUCTION

Non infectious inflammatory uveitis is a heterogeneous group of diseases, characterized by inflammation of intra-ocular structure. They can be associated with systemic autoimmune diseases, with syndromes involving several ocular structures or be a sporadic disease of unknown etiology. With an incidence of 52/100 000 person-years (1) inflammatory uveitis are responsible of 10-20% of blindness cases in developed nations. [START_REF] Wakefield | Epidemiology of uveitis[END_REF] TNFα is a cytokine that has a major role in regulating the functions of cells involved in the inflammatory process [START_REF] Caorsi | Biologic drugs in autoinflammatory syndromes[END_REF] and seems to play a key role in ocular inflammatory diseases. Indeed, the intraocular injection of TNFα to mice induces a break down of the blood-retinal barrier (4) and high levels of TNFα and TNF-receptor were observed in serum and aqueous humor of patients with uveitis. [START_REF] Rodrigues | Therapeutic monoclonal antibodies in ophthalmology[END_REF][START_REF] Lacomba | Aqueous humor and serum tumor necrosis factor-alpha in clinical uveitis[END_REF] There is an unmet need for additional effective therapies in patients with non infectious uveitis beyond corticosteroids which are the mainstay of treatment despite their well-known adverse effects. [START_REF] Jabs | Guidelines for the use of immunosuppressive drugs in patients with ocular inflammatory disorders: recommendations of an expert panel[END_REF] A better understanding of the mechanisms involved in the inflammatory response and regulation of adaptive immunity led to the development of biotherapeutics including anti-Tumor Necrosis Factor alpha (anti-TNFα). (8) A recent review of 61 studies and 1093 patients concluded that infliximab (IFX) and adalimumab (ADA) are effective in the treatment of noninfectious inflammatory uveitis with a medium level of evidence, whereas etanercept (ETA) seems to be ineffective. [START_REF] Cordero-Coma | Systematic review of anti-tumor necrosis factor-alpha therapy for treatment of immune-mediated uveitis[END_REF] Furthermore, a committee of the American expert generated guidelines for the management of biotherapies in ocular inflammatory diseases recommended the use of ADA and/or IFX as first or second line therapy according to uveitis etiologies. [START_REF] Levy-Clarke | Expert panel recommendations for the use of anti-tumor necrosis factor biologic agents in patients with ocular inflammatory disorders[END_REF] IFX and ADA were strongly recommended early in management of patients with sight threatening ocular manifestations of Behçet's disease and in second intention for children with vision-threatening uveitis secondary to JIA. [START_REF] Levy-Clarke | Expert panel recommendations for the use of anti-tumor necrosis factor biologic agents in patients with ocular inflammatory disorders[END_REF] However, data regarding factors associated with response to anti-TNFα and comparison of safety and efficacy of IFX and ADA are lacking in non infectious uveitis. In this nationwide study from the French uveitis network, our aim was to analyze and to compare the efficacy and safety of IFX and ADA in a large cohort of patients with non infectious refractory uveitis.

MATERIAL AND METHODS

Patients

Primary objective

The response to anti-TNFα and the factors associated to complete response were considered as the primary objective of this study. The response to treatment was evaluated according to the SUN Workgroup criteria. [START_REF] Jabs | Standardization of Uveitis Nomenclature (SUN) Working Group. Standardization of uveitis nomenclature for reporting clinical data. Results of the First International Workshop[END_REF] Complete response was defined as a decrease to grade 0 in level of inflammation (e.g. anterior chamber cells, vitreous haze) associated with regression of retinal vasculitis and a complete resolution of macular edema and with corticosteroids dose ≤10mg/day at 6 months. Partial response was defined as an improvement of at least 50% of inflammation and/or a significant regression of retinal vasculitis (i.e. notably asymptomatic peripheral retinal vascular leakage) and of macular edema and a reduction of >50% of initial corticosteroids dose at 6 months. All other situations were considered as non -response.

Secondary objectives

Secondary objectives included safety, corticosteroids sparing at 6 and 12 months, event free survival (e.g survival without failure, relapse and serious side effects), and comparison of clinical response, event free survival and serious side effects between IFX and ADA.

Corticosteroid sparing was assessed by comparing corticosteroid daily dose between the day of anti-TNFα introduction and after 6 and 12 months of treatment. Relapses were defined as a new ocular inflammation and/or worsening of a preexisting manifestation requiring treatment intensification. Safety was assessed by analysing the rate and type of side effects. Serious adverse events were defined as those that justified anti-TNFα treatment interruption and/or an hospitalization and/or lead to death.

Statistical analysis

Data for categorical variables are summarized as frequencies and percentages; quantitative variables are presented as the medians, 25th and 75th percentiles and were compared using Wilcoxon's rank sum test. Time to response, relapse and serious side effects were considered as outcomes in a competing risks framework; they were examined from the date of antiTNFα initiation. Partial response or end of first line antiTNFα without response were considered as competing events in the analysis of complete response; remission or treatment failure as competing events with relapse under antiTNFα; end of antiTNFα without serious side effect as competing event with the occurrence if a serious side effect. The cumulative incidences of complete response, relapse and serious side effects were estimated using Gray's method and factors associated with complete response were assessed using subdistribution hazard ratios (SHR) in Fine and Gray's models. [START_REF] Fine | A Proportional Hazards Model for the Subdistribution of a Competing Risk[END_REF] Event free survival was estimated using the Kaplan-Meier method; association of anti-TNFα with event free survival was assessed in a Cox proportional hazard model. Because of the non-randomized design, the comparison of anti-TNFα agents on complete response, serious side effect and event free survival was performed using a propensity score approach, with a matching procedure. (13) Propensity score was estimated in a logistic regression model as the probability of receiving either IFX or ADA, conditionally on baseline characteristics at the anti-TNFα introduction. A 1:1 matched sample was selected by matching IFX patients to ADA patients, within a range of 0.30 standard deviation of the logit of the estimated propensity score, without replacement. The balance in the baseline risk factors was evaluated in the matched sample by computing the standardized differences for these variables. All tests were two-sided and P-values lower than 0.05 were considered as indicating significant association. Analyses were performed using the R statistical software version 3.2.2.

RESULTS

Characteristics of the 160 patients with refractory uveitis

Two hundred and three patients with noninfectious uveitis were screened (Figure 1). Forty three were excluded because of a non-refractory uveitis or because of the use of another anti-TNFα than IFX or ADA (Figure 1). Finally, 160 patients were included of whom 98 (61%) were treated with IFX and 62 (39%) with ADA (Figure 1). Main characteristics are summarized in Table 1. The median age at anti-TNFα agent introduction was of 31 years. The median time between diagnosis of uveitis and the initiation of anti-TNFα treatment was of 51 months. Most of uveitis were bilateral (n=132; 82%), chronic (n=140; 90%), panuveitis (n=99; 62%) and 21% of them were granulomatous. Retinal vasculitis was observed in 34% of patients and macular oedema in 49% of patients. The main etiology of uveitis were Behçet's disease (BD) (n=58; 36%), juvenile idiopatic arthritis (JIA) (n=35; 22%), spondyloarthropathy (n=16; 10%), sarcoidosis (n=10; 6%), and idiopatic uveitis (n=23; 14%). The other etiologies included Vogt-Koyonagi-Harada disease (n=5), Birdshot chorioretinitis (n=6), sympathetic ophthalmia (n=2), granulomatosis with polyangitis (n=1), IRVAN syndrome (n=1), serpiginous choroiditis (n=1), systemic lupus erythematous (n=1) and relapsing polychondritis (n=1).

One hundred twenty four (84%) and 102 (64%) patients received corticosteroids and an immunosuppressant treatment, respectively, in association with anti-TNFα agent (Table 1).

The median follow-up was of 36 months after anti-TNFα initiation.

Efficacy

The cumulative incidence of overall response (complete and partial) was of 87% (CI95%: 80-91), 93% (CI95%: 87-96) and 95% (CI95%: 90-98) at 6, 12 and 24 months, respectively (Figure 2A). Among the complete responders, the median time to complete response was 2 (0-12) months. The cumulative incidence of complete response was 26% (CI95%: 19-34), 28% (CI95%: 21-36) and 29% (CI95%: 22-37) at 6, 12 and 24 months, respectively. The overall response of IFX and ADA was 97% and 95%, respectively. 2). In multivariate analysis, factors independently associated with complete response to anti-TNFα included the number of uveitis flares before anti-TNFα treatment (SHR=1.97 [1.02-3.84), p=0.045), and BD (SHR=2.52 [1.35-4.71], p=0.004).

The cumulative incidence of relapse was 7% (CI95%: 3-12), 21% (CI95%: 15-29) and 25% (CI95%: 18-33) at 6, 12 and 24 months respectively.

Corticosteroid sparing

Anti-TNFα agents had a significant corticosteroid sparing effect. The median (IQR) daily prednisolone dose was of 20 mg at time of initiation of anti-TNFα and of 10 [START_REF] Rodrigues | Therapeutic monoclonal antibodies in ophthalmology[END_REF][START_REF] Lacomba | Aqueous humor and serum tumor necrosis factor-alpha in clinical uveitis[END_REF][START_REF] Jabs | Guidelines for the use of immunosuppressive drugs in patients with ocular inflammatory disorders: recommendations of an expert panel[END_REF][START_REF] Saadoun | Biotherapies in inflammatory ocular disorders: Interferons, immunoglobulins, monoclonal antibodies[END_REF][START_REF] Cordero-Coma | Systematic review of anti-tumor necrosis factor-alpha therapy for treatment of immune-mediated uveitis[END_REF][START_REF] Levy-Clarke | Expert panel recommendations for the use of anti-tumor necrosis factor biologic agents in patients with ocular inflammatory disorders[END_REF][START_REF] Jabs | Standardization of Uveitis Nomenclature (SUN) Working Group. Standardization of uveitis nomenclature for reporting clinical data. Results of the First International Workshop[END_REF][START_REF] Fine | A Proportional Hazards Model for the Subdistribution of a Competing Risk[END_REF][START_REF] Rosenbaum | The central role of the propensity score in observational studies for causal effects[END_REF][START_REF] Suhler | A prospective trial of infliximab therapy for refractory uveitis: preliminary safety and efficacy outcomes[END_REF][START_REF] Suhler | Adalimumab therapy for refractory uveitis: results of a multicentre, open-label, prospective trial[END_REF] mg and of 7 [START_REF] Derevjanik | Quantitative assessment of the integrity of the blood-retinal barrier in mice[END_REF][START_REF] Rodrigues | Therapeutic monoclonal antibodies in ophthalmology[END_REF][START_REF] Lacomba | Aqueous humor and serum tumor necrosis factor-alpha in clinical uveitis[END_REF][START_REF] Jabs | Guidelines for the use of immunosuppressive drugs in patients with ocular inflammatory disorders: recommendations of an expert panel[END_REF][START_REF] Saadoun | Biotherapies in inflammatory ocular disorders: Interferons, immunoglobulins, monoclonal antibodies[END_REF][START_REF] Cordero-Coma | Systematic review of anti-tumor necrosis factor-alpha therapy for treatment of immune-mediated uveitis[END_REF][START_REF] Levy-Clarke | Expert panel recommendations for the use of anti-tumor necrosis factor biologic agents in patients with ocular inflammatory disorders[END_REF] mg at 6 and 12 months, respectively, (both p<0.0001 compared to baseline).

Safety

Safety related data are summarized in Table 3. Forty five (28%) patients presented at least one side effect during anti-TNFα treatment of whom twenty (12%) had at least one serious side effect. Most frequent type of side effects in patients were infection (n=18), hypersensitivity reaction (n=10), auto-immune disease (n=5) and neoplasia (n=4). The cumulative incidence of serious side events was 1% (CI95%: 0-4), 7% (CI95%: 4-12) and 8% (CI95%: 4-13) at 6, 12 and 24 months, respectively. There was a trend toward higher serious side effects with IFX (16%) compared to ADA (6%) including more infections (n=5 with IFX, n=0 with ADA), hypersensitivity reactions (n=5 with IFX, n=1 with ADA), autoimmune-diseases (n=3 with IFX, n=1 with ADA) and neoplasia (n=2 with IFX, n=1 with ADA). We analysed the factors associated with the occurrence of severe side effects with anti-TNFα. In multivariate analysis, though not significantly, incidence of severe side effects tended to be higher in older patients (SHR=1.02 [0.98-1.06]), in patients with more than 5 episodes of uveitis flare prior anti-TNFα initiation (SHR=2.76 [0.76-9.99]) and in patients with a daily corticosteroid dosage greater than 20mg (SHR=2.38 [0.69-8.20]).

Event free survival

The event free survival was of 90% (CI95%: 85-95) at 6 months, 70% (CI95%: 63-78) at 1 year and 59% (CI95%: 51-68) at 2 years (Figure 2B).

Comparison of IFX and ADA using a propensity score approach.

Eighty two patients were selected after the matching procedure 1:1. The two groups of patients (IFX vs ADA) were comparable in term of age, sex, geographic origin, uveitis characteristics, etiology, previous type of immunosuppressive treatment and associated

DISCUSSION

In this multicenter study form the French uveitis network, we analyzed a large cohort of patient treated with anti-TNFα for refractory uveitis. The most relevant messages were: 1. BD is associated with a three times higher rate of complete response to anti-TNFα therapy. 2.

Efficacy seems equivalent between IFX and ADA. 3. Trend toward higher serious adverse event was observed with IFX.

In this study, anti-TNFα agents were highly effective in the treatment of refractory uveitis with 87% and 93% of improvement at 6 and 12 months, respectively. The median time to complete response was of 2 months. A significant corticosteroïd sparing effect was evidenced with a 50% reduction of the daily dose at 6 months. These results were concordant with the published literature. Indeed, several studies reported a high rate with 68 to 82% of clinical response rate and a significant corticosteroïd sparing effect. [START_REF] Suhler | A prospective trial of infliximab therapy for refractory uveitis: preliminary safety and efficacy outcomes[END_REF][START_REF] Suhler | Adalimumab therapy for refractory uveitis: results of a multicentre, open-label, prospective trial[END_REF][START_REF] Díaz-Llopis | Treatment of refractory with adalimumab: a prospective multicenter study of 131 patients[END_REF][START_REF] Dobner | A three-centre experience with adalimumab for the treatment of non-infectious uveitis[END_REF][START_REF] Kruh | Infliximab for the treatment of refractory noninfectious Uveitis: a study of 88 patients with long-term follow-up[END_REF] We showed that BD and the number of previous uveitis flares were highly associated with complete response to anti-TNFα. The efficacy of anti-TNFα in uveitis of patients with BD was known. A study has reported an increase level of inflammatory cytokines, such as interferon γ, interleukins 2-6-17 and TNFα, in ocular fluid from uveitis patients with BD, whereas ocular fluid from patients being treated with anti-TNFα did not contain any inflammatory cytokines. They concluded that anti-TNFα agents suppress effector T cell differentiation, notably Th17 differentiation, in uveitis patients with BD. [START_REF] Sugita | Inhibition of Th17 differentiation by anti-TNF-alpha therapy in uveitis patients with Behçet's disease[END_REF] In 2011, a large literature review concerning the efficacy of anti-TNFα in BD reported an uveitis improvement in 89% and 100% of patients with IFX and ADA respectively. (20) More recently, a cohort study of about 124 uveitis patients with BD treated with anti-TNFα reported that 68% of patients were inactive at 1 year. ( 21) All these results have conducted the expert group of the Executive Committee of the American Uveitis Society to generate guidelines for the management of biotherapies in ocular inflammatory disease. They recommended the use of IFX and/or ADA early in the management of vision threatening uveitis in BD. [START_REF] Levy-Clarke | Expert panel recommendations for the use of anti-tumor necrosis factor biologic agents in patients with ocular inflammatory disorders[END_REF] In univariate analysis poorer response was observed in patients being on an immunosuppressant. However, this parameter was not independently associated with complete response to anti TNFα in multivariate analysis.

Additionally, the use of anti-TNFα is likely to be effective in active uveitis, as suggested by the association found between the number of uveitis flares before anti-TNFα and the incidence of complete response. Several studies showed an increased level of TNFα in the serum and ocular fluids of patients with active uveitis. [START_REF] Lacomba | Aqueous humor and serum tumor necrosis factor-alpha in clinical uveitis[END_REF][START_REF] Sugita | The role of soluble TNF receptors for TNF-alpha in uveitis[END_REF] Moreover, levels of TNFα seems to be correlated with recurrent uveitis. ( 6) Recently, a prospective study concluded that ADA improved significantly the visual functioning in patients with non-anterior noninfectious uveitis. [START_REF] Sheppard | Effect of Adalimumab on Visual Functioning (VFQ-25) in Visual-1 Trial Patients with Non-Anterior Non-Infectious Uveitis [abstract[END_REF] Twenty eight percent of patients presented at least one side effect, of whom 12% had to discontinue the treatment. Data concerning safety of anti-TNFα in uveitis are lacking. Our results are in line with other studies that found an overall side effect rate of 10% to 36%. [START_REF] Kruh | Infliximab for the treatment of refractory noninfectious Uveitis: a study of 88 patients with long-term follow-up[END_REF][START_REF] Arida | Anti-TNF agents for Behçet's disease: analysis of published data on 369 patients[END_REF] The most common side effects in our study were infections and hypersensitivity reaction. Bronchopneumonia, [START_REF] Arida | Anti-TNF agents for Behçet's disease: analysis of published data on 369 patients[END_REF] viral infections, [START_REF] Díaz-Llopis | Treatment of refractory with adalimumab: a prospective multicenter study of 131 patients[END_REF][START_REF] Arida | Anti-TNF agents for Behçet's disease: analysis of published data on 369 patients[END_REF] abscess, [START_REF] Arida | Anti-TNF agents for Behçet's disease: analysis of published data on 369 patients[END_REF] tuberculosis, [START_REF] Arida | Anti-TNF agents for Behçet's disease: analysis of published data on 369 patients[END_REF] cutaneous rash [START_REF] Kruh | Infliximab for the treatment of refractory noninfectious Uveitis: a study of 88 patients with long-term follow-up[END_REF] and other hypersensitivity reaction [START_REF] Arida | Anti-TNF agents for Behçet's disease: analysis of published data on 369 patients[END_REF][START_REF] Perra | Adalimumab for the treatment of Behçet's disease: experience in 19 patients[END_REF] were already described in uveitis patients treated with IFX or ADA. We also report several cases of neoplasia and autoimmune disease. A pharmacovigilance study about safety of anti-TNFα in uveitis reported an increased risk of infections, auto-immune disease and neoplasia with IFX and ADA. [START_REF] Mendes | Safety profiles of adalimumab, etanercept and infliximab: a pharmacovigilance study using a measure of disproportionality in a database of spontaneously reported adverse events[END_REF] Similar results were reported with IFX and ADA in other inflammatory diseases. [START_REF] Ramiro | Safety of synthetic and biological DMARDs: a systematic literature review informing the 2013 update of the EULAR recommendations for management of rheumatoid arthritis[END_REF][START_REF] Singh | Risk of serious infection in biological treatment of patients with rheumatoid arthritis: a systematic review and meta-analysis[END_REF][START_REF] Nanau | Safety of anti-tumor necrosis factor therapies in arthritis patients[END_REF][START_REF] Westhovens | Anti-TNF therapy and malignancy in spondyloarthritis in the Leuven spondyloarthritis biologics cohort (BIOSPAR)[END_REF] Thus, the decision of anti-TNFα treatment initiation always needs to consider the risk-benefitbalance.

In the present study, the efficacy and safety seems similar between IFX and ADA. No significant difference was found in terms of complete response or event free survival.

Few data are available regarding comparison of IFX and ADA. Two study have compared efficacy between IFX and ADA in childhood refractory chronic uveitis. [START_REF] Simonini | Prevention of flare recurrences in childhood-refractory chronic uveitis: an open-label comparative study of adalimumab versus infliximab[END_REF][START_REF] Zannin | Safety and efficacy of infliximab and adalimumab for refractory uveitis in juvenile idiopathic arthritis: 1year followup data from the Italian Registry[END_REF] The first study concluded that IFX and ADA were similar regardless of time to achieve remission and time to steroid discontinuation but ADA was superior to maintain a long term remission. [START_REF] Simonini | Prevention of flare recurrences in childhood-refractory chronic uveitis: an open-label comparative study of adalimumab versus infliximab[END_REF] The second one showed a higher overall remission rate with ADA compared to IFX. [START_REF] Zannin | Safety and efficacy of infliximab and adalimumab for refractory uveitis in juvenile idiopathic arthritis: 1year followup data from the Italian Registry[END_REF] In ulcerative colitis, a recent meta-analysis showed that IFX was more effective than ADA to induced remission but efficacy of the two anti-TNFα was comparable at one year. [START_REF] Thorlund | Adalimumab versus infliximab for the treatment of moderate to severe ulcerative colitis in adult patients naïve to anti-TNF therapy: an indirect treatment comparison meta-analysis[END_REF] Our study is, at the best of our knowledge, the first to compare the efficacy between IFX and ADA in a large cohort of non infectious uveitis. IFX and ADA seems equivalent in terms of efficacy with an overall response of 97% and 95% respectively. However, trend toward higher incidence of serious side effects was observed with IFX. Although no data are available about the safety comparison of IFX and ADA in noninfectious uveitis, two recent studies in chronic inflammatory bowel disease, including a meta-analysis, concluded that IFX and ADA presented a similar safety profile. [START_REF] Liu | Intravenous Versus Subcutaneous Anti-TNF-Alpha Agents for Crohn's Disease: A Comparison of Effectiveness and Safety[END_REF][START_REF] Mei | Infliximab is superior to other biological agents for treatment of active ulcerative colitis: A meta-analysis[END_REF] However, a trend toward a higher association between IFX and autoimmune disease and between ADA and neoplasia was found. [START_REF] Mendes | Safety profiles of adalimumab, etanercept and infliximab: a pharmacovigilance study using a measure of disproportionality in a database of spontaneously reported adverse events[END_REF] Although further studies are needed to confirm our results, this information could be interesting notably because of the different route of administration of these two agents. In fact, subcutaneous administration with ADA should be less constraining for patients, for it does not require hospitalization. By contrast, intravenous administration with IFX requires hospitalization and allows for a better monitoring of uveitis and patients overall. Moreover, the treatment cost is slightly lower for ADA than IFX (12.731 euros per year for ADA at the dose of 40 mg every 2 weeks vs 15.799 euros per year with vial optimization for IFX dosed at 5 mg/kg was administered at weeks 0, 2, and 6 and then every 8 weeks). [START_REF] Borrás-Blasco | Effectiveness of infliximab, adalimumab and golimumab for non-infectious refractory uveitis in adults[END_REF] We acknowledge some limitations in our study. Our analysis was performed as a retrospective review. We were unable to collect complete longitudinal data on patients who were seen only on an intermittent basis. Prospective enrollment and data collection from the time of diagnosis would have been ideal but is more difficult to achieve with rare diseases. Although the present study only compared these anti-TNFα agents based on observational non-randomized observations, we used a propensity score approach to minimize potential confusion bias. [START_REF] Rosenbaum | The central role of the propensity score in observational studies for causal effects[END_REF] In conclusion, anti-TNFα is highly effective in this large series of refractory inflammatory uveitis, with a similar incidence of complete response regardless of anti-TNFα agent (IFX or ADA). Behçet's disease is positively associated with complete response.

Multicenter

  retrospective observational study, from the French Uveitis Network, conducted from 2001 to 2013 and including patients with a refractory uveitis. Uveitis was considered as refractory in case of failure of at least 1 immunosuppressive and/or immunomodulator treatment, defined as fulfilling one of the two following criteria at inclusion: a. Active inflammatory chorioretinal and/or inflammatory retinal vascular lesions (fluorescein angiogram) or b. A reduction of visual acuity due to vitreous haze or macular edema (Optical Coherence Tomography, OCT). Infliximab was given intravenously at a dose of 5 mg/kg at weeks 0, 2, 6, and then every 5-6 weeks. Adalimumab (ADA) was used at the dose of 40mg every 2 weeks subcutaneously. The choice of anti TNF was left to the discretion of the physicians in charge of the patients (in France IFX and ADA can be prescribed in second line of treatment in inflammatory uveitis). The local ethics committee of Pitié Salpétrière hospital, Paris VI University, approved this study. Patients were excluded from the study if they had an infectious uveitis, if they were naïve of immunosuppressant before using anti-TNFα agent or if they were treated with other anti-TNFα than IFX or ADA. Collected data included demographic characteristics (age, sex, geographical origin), date of diagnosis, etiology of uveitis, previous treatments (corticosteroids and immunosuppressive treatment), date of anti-TNFα agent introduction, characteristic of uveitis at anti-TNFα agent introduction, indication of anti-TNFα treatment, type and dosage of anti-TNFα used in first and second line, glucocorticosteroids dose and associated treatments and outcome. Uveitis were characterized with their anatomic localization, their course (acute or chronic), the presence of granuloma, retinal vasculitis and/or macular edema according to the Standardization of Uveitis Nomenclature (SUN) Workgroup criteria. (11) All patients had a fluorescein angiogram to diagnosed retinal vasculitis and optical coherence tomography (OCT) to detect macular edema.

24 months

 24 At anti-TNFα initiation, the median visual acuity (LogMar chart) was 0.4 [0.1-1.0] for the right and 0.3 [0.0-0.8] for the left eyes. The median visual acuity improved at 6 months (0.3 [0.0-0.8] for the right and 0.1 [0.0-0.5] for the left eyes). Then visual acuity stabilized at 12 and respectively for the left eyes. In univariate analysis, the factors associated to complete response to anti-TNFα included BD (SHR=2.78 [1.55-4.99]), number of uveitis flares before anti-TNFα treatment (more than 5 flares) (SHR=1.90 [1.00-3.64]), a daily corticosteroid dosage higher than 20mg (SHR=2.10 [1.13-3.93]), and immunosuppressant treatment in association with anti-TNFα agent (SHR=0.46 [0.25-0.84]) (Table

  treatment at time of anti-TNFα initiation. The cumulative incidences of complete response and serious side effects were not significantly different regardless of anti-TNFα treatment (SHR= 0.65 [0.25-1.71], p=0.39) and 0.22 [0.04-1.25], p=0.089 respectively) (Figure 3A and 3C). The event free survival under IFX and ADA was not significantly different (SHR: 0.55, [0.28-1.08],p=0.083)(Figure3B).

Figures legends:Figure 1 :Figure 2 :Figure 3 :

 123 Figures legends:

Figure 1 :Figure 2 :Figure 3 :

 123 Figure 1: Flow chart of the study 254x190mm (300 x 300 DPI)

Table 1 : Demographic and main clinical characteristics of 160 patients with refractory uveitis

 1

	All
	n=160

Data are median

[25-

75 interquantile range], or number (percentage), aTNF: Tumor Necrosis Factor alpha antagonist, AZA: azathioprine, CYC: cyclophosphamide, MTX: methotrexate, INFγ: interferon gamma, MMF: mycophenolate mofetil, BD: Behçet disease JIA: Juvenile Idiopatic Arthritis, # 4 § 4 ‡ 11 ϴ 12 ¤ 13 † 23 ¥ 32 missing values.

Arthritis & Rheumatology

Acknowledgement :

The authors would like to thanks Emmanuel HERON, Nathalie TIEULIE, Gilles BLAISON, Yoland SCHOINDRE, Guillaume MOULIS, Thomas PAPO, Olivier LIDOVE and Jean SIBILIA for their collaboration to this study and for the management and follow-up of patients.

Contributions:

HV and DS contributed to the conception of the study. HV collected the data. HV, LB, PAB, MRR and DS analyzed the data. PS, JBF, PB, LP, BB, SA, AR, AD, DS, AP, IM, EF, EH, OF, GC, SR, JG, GP, PLH, PC, BB and DS followed the patients. HV, PS, LB, MRR, BB and DS drafted the manuscript and contributed to the writing of the article.


domain: phys sde

text:
Sylvain Petit

Solène Guitteny

Julien Robert

Pierre Bonville

Claudia Decorse

Jacques Ollivier

Hannu Mutka

Isabelle Mirebeau

Spin dynamics in highly frustrated pyrochlore magnets

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Introduction

Geometrically frustrated magnetism is a forefront research topic within condensed matter physics, as testified by the wealth of exotic phenomena discovered over the past years [START_REF]Introduction to Frustrated Magnetism[END_REF][START_REF] Gingras | [END_REF][3]. The inability of frustrated systems to satisfy all pairs of interactions simultaneously is reflected by the fact that an infinite number of different magnetic configurations minimize the classical energy, hence leading to an extensive degeneracy. This results in exotic spin dynamics, that can be studied by different means, such as magnetization, ac susceptibility, Mössbauer spectroscopy, muons and inelastic neutron scattering. For instance, quasi-elastic scattering, unraveling relaxation phenomena at short times scales as well as collective spin excitations from a given ground state (yielding the exchange interactions), or the continuum of fractionalized excitations, can be studied using state of art neutron instrumentation. Single crystal time-of-flight (TOF) data allow one to map out the reciprocal and energy space in great detail. As a result, it becomes possible to measure for instance the full dispersion of spin wave excitations, and, with the help of numerical simulations, to determine the relevant exchange constants. However, despite the high luminosity of such instruments, the counting time in one configuration (wavelength, magnetic field, temperature) is still of the order of about one day, so that measuring precisely the temperature, field dependence or the spin polarization analysis at a given Q position, remains the preserve of triple axis spectrometers (TAS). The suite of both spectrometers is thus clearly complementary. a e-mail: sylvain.petit@cea.fr It is the main objective of this paper to show these points, on the basis of recent experiments carried out on two pyrochlore magnets, namely Er 2 Ti 2 O 7 and Tb 2 Ti 2 O 7 . In Er 2 Ti 2 O 7 , a transition towards a Néel state is observed, possibly induced by an order-bydisorder mechanism. S(Q, ω) can be confronted to theory and effective exchange parameters can be extracted. Next, we examine the case of the spin liquid material Tb 2 Ti 2 O 7 . Combination of TOF and polarized triple axis measurements demonstrate the existence of a low energy propagating excitation emanating from the spin liquid ground state. Anomalies of the phonon modes, as well as of the first crystal electric field (CEF) level, suggest a strong dynamical coupling to the lattice.

Er 2 Ti 2 O 7

Since a decade, the antiferromagnetic model for XY spins on the pyrochlore lattice has been considered with much interest since it displays an extensive classical degeneracy [4][5][6] along with classical and quantum order by disorder (ObD) effects [5][6][7][8][9][10][11][12][13][14]. ObD is a very important theoretical concept in frustrated magnetism, coming into play by selecting a particular configuration out of the degenerate manifold [15,16]. At finite temperature, the selection operates because fluctuations away from this configuration allow for a relative gain of entropy (classical ObD). At zero temperature, the selection can also operate because the contribution to the total energy of zero point fluctuations is minimum for this configuration (quantum ObD). Until now, the number of confirmed examples of such a mechanism in real materials has remained scarce EPJ Web of Conferences [17]. However, Er 2 Ti 2 O 7 has recently been proposed as a most convincing case of ObD [8][9][10][11].

In Er 2 Ti 2 O 7 , the CEF acting on the Kramers Er 3+ ion is responsible for a strong XY-like anisotropy, with easy magnetic planes perpendicular to the local 111 ternary axes [START_REF] Gingras | [END_REF]6]. Er 2 Ti 2 O 7 undergoes a second order phase transition towards an antiferromagnetic noncollinear k = 0 Néel phase at T N = 1.2 K [14,[18][19][20]. The corresponding magnetic configuration, denoted ψ 2 is depicted in Fig. 1b [14,21].

The minimal model for this material takes into account an anisotropic bilinear exchange Hamiltonian written for the Er 3+ moments J i at sites i of the pyrochlore lattice (see Refs. [22] and [23]) as well as the CEF contribution, H CEF :

H = H CEF + 1 2 i, j J i • K i, j • J j , ( 1
)
where

H CEF = n,m B nm O nm is written in terms of O nm
Stevens operators [START_REF] Wybourne | Spectroscopic Properties of Rare Earths[END_REF][START_REF]The crystal field is modeled by the following coefficients: B 20 = 616 K, B 40 = 2850 K[END_REF][START_REF] Jensen | Rare Earth Magnetism[END_REF]. K i, j denotes an anisotropic exchange coupling tensor, defined in the ( a, b, c) frame attached to the Er 3+ -Er 3+ bonds [START_REF] Malkin | [END_REF]. It consists in a symmetric part with 3 parameters K a,b,c , and an antisymmetric (Dzyaloshinskii-Moriya) part with constant K 4 .

It turns out that the ground CEF state is a doublet, very well separated from the first excited states. The above model can thus be projected onto the subspace spanned by the pseudospin 1/2 that describes the two states of the ground doublet, yielding [9]:

H = i, j J zz S z i S z j -J ± S + i S - j + S - i S + j + + J ±± γ i j S + i S + j + γ * i j S - i S - j + J z± S z i ζ i j S + j + ζ * i j S - j + i ↔ j . ( 2
)
In this model, the spins are written is their local bases (ζ i j and γ i j are geometrical coefficients, see [9]) and K i, j is split in four anisotropic nearest-neighbor exchange parameters (J ±± , J ± , J z± , J zz ) with the following relations:

J zz = λ 2 z K a -2K c -4K 4 3 J ± = -λ 2 ⊥ 2K a -3K b -K c + 4K 4 12
J z± = λ ⊥ λ z K a + K c -K 4 3 √ 2
J ±± = λ 2 ⊥ 2K a + 3K b -K c + 4K 4 12
where λ ⊥,z = g ⊥,z /g J , g ⊥,z being the Landé factors.

The set of exchange parameters has been determined by inelastic neutron scattering experiments carried out in a large applied magnetic field (see right column in Table 1). In zero field, theory predicts for these parameters a quantum ObD selection of the ψ 2 state [7][8][9][10][11]. The ObD mechanism also predicts the opening of a spin gap, yet smaller that experimental findings [29][30][31]. In a recent work, the impact of the projection onto the spin 1/2 subspace was examined to see if ObD is also at play when using the full CEF Hamiltonian of Eq. ( 1). The spin excitation spectra was calculated within the Random Phase approximation [START_REF] Jensen | Rare Earth Magnetism[END_REF]32,33]. New neutron measurements were performed on a large Er 2 Ti 2 O 7 single crystal grown with the floating zone technique. The crystal was inserted in a copper sample holder and attached on the cold finger of a dilution fridge, allowing the sample to be cooled down to 50 mK. Data were collected on the IN5 time-of-flight instrument (ILL) in its single crystal set-up, with an incident neutron wavelength of 6 Å. Spin excitation spectra measured in (Q, ω) space at 50 mK are shown in Fig. 1. These new data allow one to determine a new set of anisotropic exchange parameters, see left column in Table 1. These values are more precise than those determined in Ref. [9], but are quite compatible with them, except J z± which is somewhat different. The present data also confirm the opening of a spin gap at zone centers: exp G ≈ 43 µeV [see Fig. 1c], a value which compares very well with previous estimates [29][30][31].

03012-p.2

QENS/WINS 2014

It is worth noting that the full CEF model of Eq. ( 1), which keeps a description in terms of the magnetic moments and does not perform the projection onto the pseudospin 1/2 states, also predicts the stabilization of the ψ 2 configuration. The mechanism is however different from order by disorder and the selection results rather from an energetic mechanism: an effective magnetic anisotropy is generated by the molecular field induced admixture between CEF levels. This anisotropy in turn selects the ψ 2 state and opens a spin gap in S(Q, ω). This mixing with CEF eigenstates is clearly neglected in model (2) which, by essence, considers the ground doublet only. This study thus suggests that ObD should be considered with care [34].

To conclude, the present study demonstrates that the single crystal time-of-flight measurements allow one to obtain a comprehensive dataset, measuring in "one shot" the whole spin wave dispersion. With the help of a model, quantitative information can be extracted, for instance the determination of the exchange parameters.

Tb 2 Ti 2 O 7

The pyrochlore lattice has been also widely studied for ferromagnetic interactions combined with Ising spins constrained to lie along the local z i = 111 axes ( z i link the center of each tetrahedron to its summits) [35,36]. In that case, the spins write S i = σ i z i , which, with z i . z j = - 1 3 , (i = j), leads to the following Hamiltonian:

H = i, j -J zz 3 σ i σ j = -J zz 3 i σ i 2 (3)
where denotes a tetrahedron of the structure and i runs over the summits of this tetrahedron. Since J zz ≤ 0, the minimization of the energy is ensured by the so called "ice-rule" condition i σ i = 0, stating that each tetrahedron of the pyrochlore lattice must have, in its ground-state, two spins pointing in and two spins pointing out, in close analogy with the positions of protons of water molecules in real ice [35][36][37]. There are six possible spin configurations that satisfy the ice rule on a tetrahedron yielding the same ground state degeneracy and the same entropy as in real ice [38] in the thermodynamic limit. One of the main consequences of the ice-rule is the existence of sharp and anisotropic features called "pinchpoints" that are clearly observable in diffuse neutron scattering. The spin ice case is realized in Ho 2 Ti 2 O 7 and Dy 2 Ti 2 O 7 pyrochlores where the effective ferromagnetic interaction results from the dominant dipolar interaction which overcomes the antiferromagnetic nearest neighbor exchange [39,40].

We shall now focus on other remarkable situations produced by "relaxing" the Ising nature of the spins. In this case, appreciable fluctuations between degenerate configurations are restored, resulting in a spin liquid state, a "quantum variant" of spin ices. Current theoretical descriptions are based on the generic pseudospin-1/2 Hamiltonian given by Eq. ( 2). The states of the pseudospin span the CEF ground doublet |± , the Ising exchange constant J zz is responsible for the spin-ice behavior, while the "quantum" transverse terms J ± , J z± and J ±± introduce fluctuations [9,[41][42][43][44][45].

A potential candidate for this particular regime is Tb 2 Ti 2 O 7 . It is characterized by an Ising-like anisotropy of the non-Kramers Tb 3+ ions along the local 111 axes [46,47]. In spite of effective antiferromagnetic interactions leading to a Curie-Weiss temperature of -13 K [48], which should drive the system into long-range order [32,49], prior works pointed out a disordered fluctuating ground state down to 20 mK [50,51]. Various subsequent studies have suggested complex spin dynamics, where different time and temperature scales coexist, as revealed by muons [52][53][54], magnetization [55,56] and neutron scattering experiments [57][58][59][60][61][62][63][64][65][66]. Recently, power law spin correlations have also been reported [67], bearing some resemblance with the pinch point pattern observed in the aforementioned spin ices. The peculiar form of these elastic correlations has been confirmed by recent triple axis measurements carried out at 4F (LLB), shown in Fig. 2a [68].

The observation of low energy fluctuations (in the 0-0.5 meV range), likely corresponding to fluctuations within the pseudospin-1/2 states |± , is a central issue. Recently, polarized inelastic neutron scattering experiments have been performed at IN14, that eventually demonstrate that these low energy fluctuations correspond to the superposition of a quasi-elastic response and of The scan is expected to go across the acoustic phonon dispersion. Clearly, as the Px channel shows background only, this data is an evidence that the phonon carries some magnetic intensity [78]. a propagating excitation [68]. Figures 2b andc show representative scans as well as the dispersion of this excitation within the Brillouin zone. The contribution of polarized neutrons was very important to disentangle the propagating excitation, observable in the spin-flip P z channel [blue points in the scan of Fig. 2b] from the additional quasi-elastic signal seen in the spin flip P ⊥ Q channel (green points).

It should be stressed that the existence of this low energy signal leads to very strong conclusions about the physics of Tb 2 Ti 2 O 7 : because of the intrinsic properties of non-Kramers magnetic doublets (like the Tb 3+ ground state in Tb 2 Ti 2 O 7 ), the matrix element + | J |is zero [69]. As a result, since the neutron cross section is proportional to

| + | J |-| 2 ,
no neutron signal is expected. Moreover, low energy fluctuations cannot originate from the exchange/dipole interaction. The experimental observations thus call for further interpretation. Assuming that a fluctuating molecular field mixes the ground and excited CEF states, non-zero matrix elements might in principle be restored [46,70]. However, as long as the exchange terms are one order of magnitude weaker than the gap to the first CEF excited states, the perturbed wave function should not depart significantly from |± [71,72]. This is the reason why research now keeps looking towards new physical ingredients that may directly couple the |± states. Recent studies argue that a coupling between quadrupolar moments Q may play a significant role, since + | Q |is non zero. As a result, the degeneracy of the |± states should be lifted, providing a first explanation for the low energy dispersing signal. A model based on the most simple on-site quadrupolar term has been proposed, phenomenologically connected with a possible tetragonal distortion precursor to a T 0 Jahn-Teller transition [53,61,[70][71][72][75][76][77]. Despite being rather successful in explaining a number of experimental results [61,[71][72][73][74], this model is however incomplete as it predicts a CEF singlet state on each site, a feature which is not compatible with the existence of elastic correlations.

At slightly larger energies, ω ∼ 1 -2 meV, the inelastic response is dominated by the first CEF excitation located at ≈ 1.5 meV [see Figs. 2d ande]. The line shape of this CEF excitation is much more complicated than a single dispersionless mode and very likely contains several excitation modes. The line shape is modulated already at 10 K and down to the base temperature of 50 mK, because of the interactions between Tb 3+ magnetic moments [32]. In a very narrow range of scattering vectors Q close to crystalline zone centers, such as (1, 1, 1) and (2, 2, 0), an unexpected upturn of the dispersion is observed [see Figs. 2d ande]. This upturn arises in the region of reciprocal space where a crossing occurs between the crystal field line and the acoustic phonon branch stemming from the zone centers. Very recently, Tom Fennell et al have shown that this phonon is best described by a magneto-elastic mode [78]. The authors argue that in the spin liquid state, it is coupled to the excited crystal field level forming a hybrid excitation QENS/WINS 2014 and suggesting that Tb 2 Ti 2 O 7 is a "magnetoelastic spin liquid" (yet the hybridization mechanism remains to be further elucidated). To illustrate this point, Fig. 2f shows a cut along h, h, 0 , across the dispersion of the phonon mode, taken at an energy transfer of 1.8 meV. This experiment was carried out with polarized neutron analysis, in collaboration with the authors of Ref. [78]. In the P Q configuration, the non-spin-flip signal shows no intensity while it actually does in the P ⊥ Q and P z configurations. Clearly, the conclusion is that the phonon carries some magnetic intensity. The magneto-elastic coupling responsible for the phonon and CEF anomalies could be the driving force leading to the wanted effective interactions between quadrupoles. Indeed, this magneto-elastic coupling would involve in the Hamiltonian a term of the form u.Q, where u denotes the phonon displacement operator. Integrating out the phonon variables could then result in an effective interaction between quadrupoles mediated by phonons [79][80][81]. Note that there are additional clues in favor of a strong dynamical spin-lattice coupling: structural fluctuations below 15 K observed by high resolution X diffraction [82], giant magneto-striction [83] and the instability of the spin liquid state versus pressure and stress [84], all of which have been reported recently, but no static distortion has been observed so far [85].

Conclusion

To conclude, single crystal time-of-flight measurements now yield very detailed data, which can be confronted to spin dynamics simulations. Measurements in variable conditions of pressure, magnetic field, temperature, and especially polarized techniques continue however to be the preserve of triple axis spectrometers. In this respect both techniques are fully complementary.

In the context of pyrochlores, the study of the spin dynamics in Er 2 Ti 2 O 7 demonstrates that quantitative information can be extracted from single crystal TOF data, provided models are available. This supposes a close connection with theoreticians. In Tb 2 Ti 2 O 7 , the combination of TOF and polarized triple axis measurements demonstrate the existence of a low energy propagating excitation emanating from the spin liquid ground state. Anomalies of the phonon modes, as well as of the first CEF level, suggest a strong dynamical coupling to the lattice.

Figure 1 .

 1 Figure 1. a) Sketch of the pyrochlore lattice. Rare earth ions only are shown for sake of clarity. b) Sketch of the ψ 2 configuration in Er 2 Ti 2 O 7 . c) INS data recorded at IN5, showing cuts in (Q, ω) space in the ordered phase of Er 2 Ti 2 O 7 . d) IN5 raw data recorded with an incident wavelength of 8.5 Å at Q = (111) and T = 50 mK showing the spin gap (blue line) at 43 µeV.

Figure 2 .

 2 Figure 2. a) Elastic scattering map taken in Tb 2 Ti 2 O 7 at 70 mK. Symmetrization has been performed for clarity of the figure. Pinch points are observed at Q = (111) for instance, bearing resemblance with the situation in classical spin ices. b) IN14 polarized raw data taken at 50 mK in the spin flip channel for different polarizations (Px in red, Py in green and Pz in blue). The propagating character is detected in the Pz channel. c) Dispersion of the propagating excitation in Tb 2 Ti 2 O 7 from Ref. [68]. The studied Q directions are those shown by red lines on a). d) and e) are respectively 4F (triple axis at LLB) and IN5 (TOF at ILL) raw data showing the modulation of the first excited CEF level around 1.5 meV at base temperature. f) Constant energy scans along the dotted line in e) for different non spin flip polarizations.The scan is expected to go across the acoustic phonon dispersion. Clearly, as the Px channel shows background only, this data is an evidence that the phonon carries some magnetic intensity[78].

Table 1 .

 1 Anisotropic exchange parameters in Er 2 Ti 2 O 7 . Units are in 10 -2 meV. Positive values correspond to AF interactions.

	Coupling Fit based on Hamiltonian (1)	Ref. [9]
	J ±±	4.54 (±0.1)	4.2 (±0.5)
	J ±	5.84 (±0.1)	6.5 (±0.75)
	J z±	0.92 (±0.1)	-0.88 (±1.5)
	J zz	-0.87 (±0.1)	-2.5 (±1.8)

We acknowledge E. Lhotel, L. Jaubert, P. McClarty, Michel Gingras and Jeff Rau for fruitful discussions. We also thank S. Turc and P. Boutrouille (cryogeny group at ILL and LLB respectively) for their technical help while setting up the magnet and the dilution fridge.


Nicolás Espitia
email: nicolas.espitia-hoyos@gipsa-lab.fr

Antoine Girard
email: antoine.girard@l2s.centralesupelec.fr

Nicolas Marchand
email: nicolas.marchand@gipsa-lab.fr

Christophe Prieur
email: christophe.prieur@gipsa-lab.fr

Event-based stabilization of linear systems of conservation laws using a dynamic triggering condition

Keywords: Event-based control, hyperbolic systems, Lyapunov techniques, triggering conditions, piecewise continuous functions

). The well-posedness as well as the global exponential stability of the resulting closed-loop system is studied. Some numerical simulations are performed to validate the theoretical results.

INTRODUCTION

In recent years, event-based control has gained a lot of attention not only because of its efficient way of using communications and computational resources by updating control inputs aperiodically (only when needed) but also because of its rigorous way to implement digitally continuous time controllers. For finite dimensional networked control systems, event-triggered strategies for stabilization have become an active research area, for which seminal contributions can be found in [START_REF] Åström | Comparison of periodic and event based sampling for first-order stochastic systems[END_REF]; Årzén (1999) or more recent ones in [START_REF] Heemels | An introduction to event-triggered and self-triggered control[END_REF]; [START_REF] Marchand | A general formula for event-based stabilization of nonlinear systems[END_REF]; [START_REF] Postoyan | A framework for the event-triggered stabilization of nonlinear systems[END_REF] and the references therein. Typically, the framework of eventbased control includes a feedback control law which is designed to stabilize the system along with a triggering strategy which determines the time instants when the control needs to be updated. The triggering strategy guarantees that a Lyapunov function decreases strictly. The most common triggering strategy uses a static rule obtained by an Input-to-State Stability (ISS) property as in [START_REF] Tabuada | Event-triggered real-time scheduling of stabilizing control tasks[END_REF]. An extension to this strategy is done in [START_REF] Girard | Dynamic triggering mechanisms for event-triggered control[END_REF] where an internal dynamics is introduced into the triggering rule, reducing the number of control updates in comparison to the static policy. Other approaches, among others, rely directly on the time derivative of the Lyapunov function [START_REF] Marchand | A general formula for event-based stabilization of nonlinear systems[END_REF]; [START_REF] Seuret | Stability of non-linear systems by means of event-triggered sampling algorithms[END_REF]).

The design of event-based control strategies for infinite dimensional systems (namely those governed by partial differential equations (PDEs)), is rarely treated in the literature. For parabolic PDEs, event-based strategies are considered in [START_REF] Selivanov | Distributed eventtriggered control of transport-reaction systems[END_REF]. For a class of hyperbolic systems of conservation laws, the closest framework to event-based control is the work on switched hyperbolic systems as in [START_REF] Lamare | Switching rules for stabilization of linear systems of conservation laws[END_REF] which is highly inspiring, especially when dealing with the wellposedness of the closed-loop solution and with the filter mechanism in form of a dynamic variable enabling to reduce the number of switches. A recent work however has introduced two event-based boundary controllers for linear hyperbolic systems of conservations laws: inspired by two of the main strategies developed for finite dimensional systems, an extension by means of Lyapunov techniques for stability has been done in [START_REF] Espitia | Event-based control of linear hyperbolic systems of conservation laws[END_REF] for linear hyperbolic systems of conservation laws. It is worth recalling that stability analysis and continuous stabilization of such systems by means of boundary control have been considered for a long time in literature. For instance, backstepping design [START_REF] Krstic | Backstepping boundary control for first-order hyperbolic PDEs and application to systems with actuator and sensor delays[END_REF]) and Lyapunov techniques [START_REF] Coron | A strict Lyapunov function for boundary control of hyperbolic systems of conservation laws[END_REF]) are the most commonly used. In fact, some complex physical networks can be modeled by means of Hyperbolic PDEs. To mention few applications which stand out: hydraulic [START_REF] Bastin | Using hyperbolic systems of balance laws for modeling, control and stability analysis of physical networks[END_REF]), road traffic [START_REF] Coclite | Traffic flow on a road network[END_REF]), gas pipeline networks [START_REF] Gugat | Gas flow in fan-shaped networks: Classical solutions and feedback stabilization[END_REF]. They all motivate the use of boundary control. Furthermore, they all motivate the event-based boundary control which is actually a realistic approach for the actuator in those systems. In order to make the motivation a bit more clear, for instance in open channels modeled by the Saint-Venant equations, the actuation on the boundary might be expensive due to the actuator inertia when regulating the water level and the water flow rate by using gates opening as a control actions. Event-based control would suggest to modulate efficiently the gates opening, only when needed. The main contribution of this work relies on the extension of one of the event-based strategies proposed in [START_REF] Espitia | Event-based control of linear hyperbolic systems of conservation laws[END_REF]. We introduce an internal dynamic to the triggering algorithm in order to reduce the number of control updates while guaranteeing both the well-posedness of the closedloop solution and the global exponential stability as well as the absence of the so-called Zeno phenomena. This paper is organized as follows. Section 2 contains some results provided in [START_REF] Espitia | Event-based control of linear hyperbolic systems of conservation laws[END_REF]. The main result of this paper is then presented in Subsection 2.3. Section 3 provides a numerical example to illustrate the main results and to compare the two control strategies for the control of a system describing traffic flow on a roundabout. Finally, conclusions are given in Section 4.

Preliminary definitions and notation. The set of all functions φ :

[0, 1] → R n such that 1 0 |φ(x)| 2 < ∞ is de- noted by L 2 ([0, 1], R n ) that is equipped with the norm • L 2 ([0,1],R n ) .
The restriction of a function y : I → J on an open interval (x 1 , x 2 ) ⊂ I is denoted by y| (x1,x2) . Given an interval I ⊆ R and a set J ⊆ R n for some n ≥ 1, a piecewise left-continuous function (resp. a piecewise right-continuous function) y : I → J is a function continuous on each closed interval subset of I except maybe on a finite number of points x 0 < x 1 < . . . < x p such that for all l ∈ {0, .., p -1} there exists y l continuous on [x l , x l+1 ] and y l|(x l ,x l+1 ) = y |(x l ,x l+1 ) . Moreover, at the points x 0 , • • • , x p the function is continuous from the left (resp. from the right). The set of all piecewise left-continuous functions (resp. piecewise right-continuous functions) is denoted by C lpw (I, J) (resp. C rpw (I, J)). In addition, we have the following inclusions

C lpw ([0, 1], R n ), C rpw ([0, 1], R n ) ⊂ L 2 ([0, 1], R n ).
Linear Hyperbolic Systems Let us consider the linear system of conservation laws given in Riemann coordinates:

∂ t y(t, x) + Λ∂ x y(t, x) = 0 x ∈ [0, 1], t ∈ R + (1)
×[0, 1] → R n , Λ is a diagonal matrix in R n×n such that Λ = diag(λ 1 , • • • , λ n ) with 0 < λ 1 < λ 2 < • • • < λ n , H ∈ R n×n , B ∈ R n×m and u : R + → R m .
In addition, we consider the initial condition given by y(0, x) = y 0 (x),

x ∈ [0, 1] (3) where y 0 ∈ C lpw ([0, 1], R n ). We assume that the linear hyperbolic system is only observed at right boundary x = 1 at any time. Therefore we define the output function as follows:

z(t) = y(t, 1) (4)
2. EVENT-BASED STABILIZATION

Preliminaries on stability

We define the notion of stability considered in the paper. Definition 1. The linear hyperbolic system (1)-( 3),( 4) with controller u = ϕ(z) is globally exponentially stable (GES) if there exist ν > 0 and C > 0 such that, for every

y 0 ∈ C lpw ([0, 1]; R n ), the solution satisfies, for all t in R + , y(t, •) L 2 ([0,1];R n ) ≤ Ce -νt y 0 L 2 ([0,1];R n ) (5)
A particular case studied in literature (see e.g. [START_REF] De Halleux | Boundary feedback control in networks of open channels[END_REF]) is when ϕ is given by u = ϕ c (z) as u(t) = Kz(t). This corresponds to continuous time control for which it holds y(t, 0) = Gz(t) t ∈ R + (6) with G = H + BK. The following inequality is stated in [START_REF] Coron | Dissipative boundary conditions for one-dimensional nonlinear hyperbolic systems[END_REF] as a sufficient condition, usually called dissipative boundary condition, which guarantees that the system (1)-( 3) with boundary condition ( 6) is GES. In this paper, such a sufficient condition is assumed to be satisfied. Assumption 1. The following inequality holds:

ρ 1 (G) = Inf ∆G∆ -1 ; ∆ ∈ D n,+ < 1 (7)
where • denotes the usual 2-norm of matrices in R n×n and D n,+ denotes the set of diagonal matrices whose elements on the diagonal are strictly positive. Proposition 1. [START_REF] Diagne | Lyapunov exponential stability of 1-D linear hyperbolic systems of balance laws[END_REF]] Under Assumption 1, there exist µ > 0, and a diagonal positive definite matrix

Q ∈ R n×n (with Q = Λ -1 ∆ 2
) such that the following matrix inequality holds G T QΛG < e -2µ QΛ.

(8) Moreover, the linear hyperbolic system (1)-( 3),( 4),( 6) is GES and (5) holds for some C > 0 and ν = µλ where λ = min 1≤i≤n {λ i }.

Under the assumption of Proposition 1, inspired by (Diagne et al., 2012, Theorem 1), let us recall that the function defined, for all y

(•) ∈ L 2 ([0, 1], R n ), by V (y) = 1 0 y(x) T Qy(x)e -2µx dx (9)
is a Lyapunov function for system (1)-( 3),( 4),(6).

ISS static event-based stabilization

We introduce in this subsection the main results of one event-based control scheme for linear hyperbolic systems of conservation laws introduced in [START_REF] Espitia | Event-based control of linear hyperbolic systems of conservation laws[END_REF]. In that framework, ISS property with respect to a deviation between the continuous controller and the event-based controller, combined with a strict Lyapunov condition using (9), has been studied.

Definition 2. [Definition of ϕ s ] Let ς, κ, η, µ > 0, K in R m×n , Q a diagonal positive matrix in R n×n .
Let us define ϕ s the operator which maps z to u as follows:

Let z be in C rpw (R + , R n ) and let Ṽ be given, at t = 1 λ , by

Ṽ ( 1 λ ) = n i=1 Q ii 1 0 H i z(t -x λi ) 2 e -2µx dx (10)
and, for all t > 1 λ , by

Ṽ (t) = n i=1 Q ii 1 0 H i z(t -x λ i ) + B i u(t -x λ i ) 2 e -2µx dx (11) and let ε(t) = ς Ṽ ( 1 λ )e -ηt for all t ≥ 1 λ . If Ṽ ( 1 λ ) > 0, let the increasing sequence of time instants (t u k ) be defined iteratively by t u 0 = 0, t u 1 = 1 λ , and for all k ≥ 1, t u k+1 = inf{t ∈ R + |t > t u k ∧ BK(-z(t) + z(t u k )) 2 ≥ κ Ṽ (t) + ε(t)} (12) If Ṽ ( 1 λ ) = 0, the time instants are t u 0 = 0, t u 1 = 1 λ and t u 2 = ∞.
Finally, let the control function, z → ϕ s (z)(t) = u(t), be defined by:

u(t) = 0 ∀t ∈ [t u 0 , t u 1 ) u(t) = Kz(t u k ) ∀t ∈ [t u k , t u k+1 ), k ≥ 1 (13)
Remark 1. The boundary condition (2) with controller u = ϕ s (z) as defined in Definition 2 can be rewritten as:

y(t, 0) = Gz(t) + d(t) t ∈ R + (14) where d(t) = BK(-z(t) + z(t u k )) t ∈ [t u k , t u k+1 ) (15
) which can be seen as a deviation between the continuous controller u = Kz and the event based controller of Definition 2.

• Proposition 2. [START_REF] Espitia | Event-based control of linear hyperbolic systems of conservation laws[END_REF]] Let y be a solution to (1)-( 3). It holds that, for all t ≥ 1 λ , V (y(t, •)) = Ṽ (t), where Ṽ (t) is given by (11). Theorem 1. [START_REF] Espitia | Event-based control of linear hyperbolic systems of conservation laws[END_REF]] Let K be in R n×n such that Assumption 1 holds for G = H +BK. Let µ > 0, Q a diagonal positive matrix in R n×n and ν = µλ be as in Proposition 1. Let σ be in (0, 1), α > 0 such that

(1 + α)G T QΛG ≤ e -2µ QΛ. Let ρ be the largest eigenvalue of (1 + 1 α )QΛ, κ = 2νσ ρ , η > 2ν(1 -σ)
and ε and ϕ s be given in Definition 2. Let V be given by (9). Then the system (1)-( 3),( 4) with the controller u = ϕ s (z) has a unique solution and is globally exponentially stable.

ISS dynamic event-based stabilization

In this section we introduce a second event-based control strategy relying on the previous one. It is inspired by [START_REF] Girard | Dynamic triggering mechanisms for event-triggered control[END_REF] (for finite dimensional systems) where an internal dynamic variable is added to the event triggering condition in order to reduce the number of triggering times while guaranteeing the exponential stability. We recall that in ISS static event-based stabilization, events are triggered so that d 2 -κ Ṽ is always less than ε (see ( 12)). In this new approach, we will rather impose that the weighted average value of d 2 -κ Ṽ -ε is less than 0. Then, an internal dynamic will be presented under the form

m(t) = e -ηt t 1 λ e ηs -κ Ṽ (s) -ε(s) + d(s) 2 ds for all t ≥ 1 λ . Definition 3. [Definition of ϕ d ] Let σ be in (0, 1), Ṽ (t), ε(t)
given as in Definition 2 for all t ≥ 1 λ , and ρ and κ as in Theorem 1. Let us define ϕ d the operator which maps z to u as follows:

Let z be in C rpw (R + , R n ). If Ṽ ( 1 λ ) > 0
, let the increasing sequence of time instants (t u k ) be defined iteratively by t u 0 = 0, t u 1 = 1 λ , and for all k ≥ 1,

t u k+1 = inf{t ∈ R + |t > t u k ∧ m(t) ≥ 0} (16) where m satisfies the differential equation, ṁ(t) = -ηm(t) + (-κ Ṽ (t) -ε(t) + BK(-z(t) + z(t u k )) 2 ) m( 1 λ ) = 0 (17) for all t ∈ [t u k , t u k+1 ) for a given η > 2ν(1 -σ).
If Ṽ ( 1 λ ) = 0, the time instants are t u 0 = 0, t u 1 = 1 λ and t u 2 = ∞. Finally, let the control function, z → ϕ d (z)(t) = u(t), be defined by:

u(t) = 0 ∀t ∈ [t u 0 , t u 1 ) u(t) = Kz(t u k ) ∀t ∈ [t u k , t u k+1 ), k ≥ 1 (18)
Note that m(t u k ) = 0 for all k ≥ 1. Proposition 3. For any y 0 in C lpw ([0, 1], R n ), there exists an unique solution to the closed-loop system (1)-( 3),( 4) and controller u = ϕ d (z).

Proof. We recall a sufficient condition for the existence and uniqueness of solutions under a feedback control. Lemma 1. [START_REF] Espitia | Event-based control of linear hyperbolic systems of conservation laws[END_REF]] Let ϕ be an operator from

C rpw (R + , R n ) to C rpw (R + , R m ) satisfying the follow- ing causality property: for all s in R + , for all z, z * ∈ C rpw (R + , R n ) (∀t ∈ [0, s], z(t) = z * (t)) =⇒ (∀t ∈ [0, s], u(t) = u * (t)) where u = ϕ(z) and u * = ϕ(z * ). Let y 0 ∈ C lpw ([0, 1], R n ).
Then, there exists a unique solution to the closed-loop system (1)-( 3) with controller u = ϕ(z) where z is defined by (4). Moreover, for all

t in R + , y(t, •) ∈ C lpw ([0, 1], R n ) and for all x ∈ [0, 1] y(•, x) ∈ C rpw (R + , R n ).
We will show that ϕ d defined in Definition 3 satisfies hypothesis of Lemma 1. Once it is done, the result of Proposition 3 yields with ϕ d .

Let us then prove that

u = ϕ d (z) belongs to C rpw (R + , R m ) provided z is in C rpw (R + , R n ). Consider J a closed interval subset of R + . Since z is in C rpw (R + , R n ),
z has a finite number of discontinuities on J. We denote t z 1 , • • • , t z M ∈ J as the increasing sequence of these discontinuity time instants to which we add the extremities t z 0 and t z M +1 of the interval J. The goal is to prove that u has a finite number of discontinuities on the time interval [t z i , t z i+1 ], with i ∈ {0, ..., M }. If Ṽ ( 1 λ ) = 0, there is only at most one discontinuity which is t u 1 = 1 λ . Let us see the case Ṽ ( 1 λ ) > 0. We define w i (t) as the continuation of BKz(t) on the interval [t z i , t z i+1 ] with the left limit of BKz(t) in t z i+1 , that is

w i (t) = BKz(t), if t ∈ [t z i , t z i+1 ) (19) w i (t z i+1 ) = lim t→(t z i+1 ) -BKz(t) (20)
The definition of C rpw (R + , R n ) insures that the left limit of BKz(t) exists and that w i (t) is continuous on the closed interval [t z i , t z i+1 ]. Then, it is uniformly continuous. It means that for all ζ > 0, there exists τ > 0 such that

∀t, t ∈ [t z i , t z i+1 ] : |t -t | < τ ⇒ w i (t) -w i (t ) 2 < ζ We denote τ the value of τ when ζ = ε(t z i+1
). We assume that there are at least two consecutive discontinuity instants in (t z i , t z i+1 ) and let t u k be the first one. Considering ( 16) and ( 17) in Definition 3 and using the continuity of m, ε and w i , it holds at time t = t u k+1 : m(t u k+1 ) ≥ 0 (21) Let us prove by contradiction that |t u k -t u k+1 | ≥ τ . To do that, let us assume that |t u k -t u k+1 | < τ . Then, by uniform continuity, we have

w i (t u k ) -w i (s) 2 < ε(t z i+1 ) for all s ∈ [t u k , t u k+1 ].
Since ε is a decreasing function, it holds also that w i (t u k ) -w i (s) 2 < ε(s). Due to the non-negativity of Ṽ , we have

w i (t u k ) -w i (s) 2 < ε(s) + κ Ṽ (s) (22)
Multiplying both sides of ( 22) by e ηs and integrating on [t u k , t u k+1 ], it yields,

t u k+1 t u k e ηs w i (t u k ) -w i (s) 2 ds < t u k+1 t u k e ηs ε(s)ds + t u k+1 t u k e ηs κ Ṽ (s)ds
Multiplying both sides by e -ηt u k+1 and re-organizing the previous inequality, one gets, 23) is equivalent to m(t u k+1 ) < 0. which contradicts (21). Hence, |t u k -t u k+1 | ≥ τ . Therefore, τ gives a lower bound for the duration between two input updates, depending only on the interval (t z i , t z i+1 ). Finally, an upper bound for the maximal number of input updates on (t z i , t z i+1 ) is given by:

e -ηt u k+1 t u k+1 t u k e ηs ( w i (t u k ) -w i (s) 2 -κ Ṽ (s) -ε(s))ds < 0 (23) Using (17) and m(t u k ) = 0, for all t ≥ t u k , we have that m(t) = e -ηt t t u k e ηs -κ Ṽ (s) -ε(s) + w i (t u k ) - w i (s) 2 ds Then, (
s i = t z i+1 -t z i τ
The number of discontinuities of u on J is bounded by S = M i=1 s i + M + 2 which is finite. To conclude, from (18) in Definition 3, u is piecewise constant, which yields u ∈ C rpw (R + , R m ). Due to the limitation of space, we refer to [START_REF] Espitia | Event-based control of linear hyperbolic systems of conservation laws[END_REF] to see the type of arguments that need to be used for the proof that operator ϕ d satisfies the causality property.

We conclude then that Lemma 1 holds. This ends the proof of Proposition 3.

•

Let us now state our main result of the paper. Theorem 2. Let K be in R n×n such that Assumption 1 holds for G = H + BK. Let µ > 0, Q a diagonal positive matrix in R n×n and ν = µλ be as in Proposition 1. Let σ be in (0, 1), η and ε and ϕ d be given in Definition 3. Let V be given by ( 9) and d given by (15). Then the system (1)-( 3),( 4) with the controller u = ϕ d (z) has a unique solution and is globally exponentially stable.

Proof. The existence and uniqueness of a solution to system (1)-( 3),(4), with u = ϕ d (z) is given by Proposition 3. Let us show that the system is GES but before proceeding, it is important to recall the following lemma which will be necessary to show that the system is GES and whose proof is given in the Appendix in [START_REF] Espitia | Event-based control of linear hyperbolic systems of conservation laws[END_REF]. Lemma 2. Let y be a solution to (1)-( 3) and let V (y) be given by (9). Then, t → V (y(t, •)) is continuous and right differentiable on R + and its right time-derivative (denoted by D + V ) is given by:

D + V = y T (•, 0)QΛy(•, 0) -y T (•, 1)e -2µ QΛy(•, 1) -2µ 1 0 y T (Λe -2µx Q)ydx (24)
Having stated this, assume first that Ṽ ( 1 λ ) > 0. Thanks to the boundary condition (2) with u = ϕ d (z), we obtain from its equivalent form ( 14) that ( 24) can be rewritten as follows: 25) Using Young's inequality and the fact that (1 + α)G T QΛG ≤ e -2µ QΛ, then from (25) it follows:

D + V = (Gz) T QΛGz + 2(Gz) T QΛd + d T QΛd -z T e -2µ QΛz -2µ 1 0 y T (e -2µx ΛQ)ydx (
D + V ≤ -2µ 1 0 y T ΛQye -2µx dx + (1 + 1 α )d T QΛd Since Q is diagonal positive definite, it holds ΛQ ≥ λQ. Thus, taking ν = µλ, it yields, D + V ≤ -2νV + (1 + 1 α )d T QΛd
which can be rewritten as follows:

D + V ≤ -2νV + ρ d 2 (
26) To show the global exponential stability of the closed-loop system, we consider the following Lyapunov function candidate W , for the augmented dynamical system, defined, for all y(•)

∈ C lpw ([0, 1], R n ) and m ∈ R -, ε ∈ R + , by W (y, m, ε) = V (y) + ρ η-2ν(1-σ) ε -ρm (27)
Computing the right time-derivative of ( 27), it yields,

D + W = D + V -η ρ η-2ν(1-σ) ε -ρ(-ηm -κ Ṽ -ε + d 2 ) ( 28
) Then, replacing (26) in (28), using κ = 2σν ρ and applying Proposition 2, we obtain for all t ≥ 1 λ ,

D + W (t) ≤ -2ν(1 -σ)V (t) +ρηm(t) + ρε(t) -η ρ η-2ν(1-σ) ε(t)
which can be rewritten as follows:

D + W (t) ≤ -2ν(1 -σ)(W (t) - ρ η-2ν(1-σ) ε(t) + ρm(t)) +ρηm(t) + ρε(t) -η ρ η-2ν(1-σ) ε(t)
Simplifying the previous inequality, one gets

D + W (t) ≤ -2ν(1 -σ)W (t) + ρ(-2ν(1 -σ) + η)m(t)
From the definition of ϕ d , events are triggered in order to guarantee for all t ≥ 1 λ , that m(t) ≤ 0. We obtain accordingly, for all t ≥ 1 λ , D + W (t) ≤ -2ν(1 -σ)W (t) Now, using the Comparison principle, for all t ≥ 1 λ , we have

V (y(t, •)) ≤ W (y(t, •), m, ε) ≤ e -2ν(1-σ)(t-1 λ ) W (y( 1 λ , •), m, ε) (29)
The previous inequality holds even if Ṽ ( 1 λ ) = 0 since in this case W (y( 1 λ , •), m, ε) = 0 for all t ≥ 1 λ . Knowing that

m( 1 λ ) = 0 and ε( 1 λ ) = ςV (y( 1 λ , •))e -η 1
λ , inequality (29) can be rewritten as follows,

V (y(t, •)) ≤ e -2ν(1-σ)(t-1 λ ) V (y( 1 λ , •)) + ρς η-2ν(1-σ) V (y( 1 λ , •))e -η 1 λ (30)
In addition, V (y( 1 λ , •)) is given as follows (see [START_REF] Espitia | Event-based control of linear hyperbolic systems of conservation laws[END_REF] for further details):

V (y( 1 λ , •)) ≤ e 2θ λ λ e 2(θ+µ) V (y 0 ) (31)
Therefore, replacing ( 31) in ( 30) we get for all t ≥ 1 λ ,

V (y(t, •)) ≤ e 2θ λ λ +2(θ+µ) ×   1 + ρςe -η 1 λ η -2ν(1 -σ)   e -2ν(1-σ)t V (y 0 )
This ends the proof of Theorem 2.

•

The following proposition states that the first triggering time after t = 1 λ occurs with ϕ s than with ϕ d . (Its proof is omitted due to space limitation). Proposition 4. Let t u 2,s be given by the rule (12) and let t u 2,d be given by the rule (16). It holds that after t = 1 λ , t u 2,s ≤ t u 2,d .

NUMERICAL SIMULATIONS

We illustrate our results by considering the following example of a linear system of 2×2 hyperbolic conservation laws describing traffic flow on a simple roundabout (see [START_REF] Espitia | Event-based control of linear hyperbolic systems of conservation laws[END_REF]):

∂ t y + Λ∂ x y = 0 ( 32
)
with y = [y 1 y 2 ] T and Λ = diag(1 √ 2), the boundary condition given by y(t, 0) = Hy(t, 1) + Bu(t) where H = ( 0 0.7 0.9 0 ), B = I 2 and u(t) = Ky(t, 1) with K = 0 k1 k2 0 . The initial condition is y(0, x) = [ 4x(x-1) sin(8πx) ]

T for all x ∈ [0, 1].

Continuous stabilization: controller

u = ϕ c (z) Here, u(t) = ϕ c (z)(t) = Kz(t) is the continuous controller acting from t ≥ 1 λ = 1. K has been designed such that ρ 1 (G) < 1 with G = H + BK. With K = 0 0.3 -0.9 0 and ∆ G = ( 0.9134 0 0 1.2580 ), ∆ G G∆ -1 G = 0.7262 < 1.
It implies that the closed-loop system is GES. Condition (8) in Proposition 1 was checked with scalars µ = 0.1, ν = 0.1 and the symmetric matrix Q = ( 0.8346 0 0 1.1191 ).

ISS static event-based stabilization: controller

u = ϕ s (z)
The boundary condition is now y(t, 0) = Hy(t, 1) + Bu(t) where u(t) = ϕ s (z)(t). The parameters for the triggering algorithm are α = 0.5, σ = 0.9. Therefore, ρ = 4.7481, κ = 0.0379 and [(1 + α)G T QΛG -e -2µ QΛ] = -0.6833 0 0 -0.0439 is a symmetric negative definite matrix. Consequently, 12) is chosen to be ε(t) = ςV (1)e -ηt , t ∈ R + with η = 1, V (1) = 0.6390 and ς is such that ςV (1) = 1 × 10 -2 . The number of events under this event-based approach was 109, counting them from t ≥ 1 λ = 1.

ISS dynamic event-based stabilization: controller

u = ϕ d (z)
The boundary condition is now y(t, 0) = Hy(t, 1) + Bu(t) where u(t) = ϕ d (z)(t). The number of events under this event-based approach was 86, counting them from t ≥ 1.

Figure 1 shows functions V when stabilizing with ϕ s and ϕ d . It can be noticed that under the two eventbased controllers ϕ s and ϕ d , global asymptotic stability is achieved with quite different observed rates despite similar theoretical guarantees. Besides this, the first triggering time occurs with ϕ s . This is consistent with Proposition 4. In addition, for both event-based approaches, we ran simulations for several initial conditions given by y 0 a,b (x) = [ax(1 -x) b 2 sin((2a)πx)] T , a = 1, ..., 5 and b = 1, ..., 10 on a frame of 8 s. We have computed the duration intervals between two control updates (inter-execution times). The mean value, standard deviation and the coefficient of variation of inter-execution times for both approaches are reported in Table 1 and the density of such inter-execution times is given in Figure 2. From this figure and Table 1, it can be observed that stabilization with ϕ d results in larger inter-execution times than with ϕ s which was expected because events generated according to ϕ d -event-triggered rule, is a weighted average of those generated according to ϕ s -event-triggered rule. The mean value of triggering times with ϕ s was 158.3 events whereas, with ϕ d , it was 109.1 events. It can be seen that using ϕ d results in larger inter-execution times in average than ϕ s . In addition, ϕ d Table 1. Mean value, standard deviation and variability of inter-execution times for ϕ s and ϕ d . reduces the variability of the inter execution times and with ϕ s it is needed to sample faster than with ϕ d .

CONCLUSION

In this paper, a new event-based boundary controller has been proposed. The analysis of global exponential stability is based on Lyapunov techniques. We have proved that under the new event-based stabilization strategy, the solution to the closed-loop system exists and is unique. This work leaves some open questions for future works.

The event-based stabilization approaches may be applied to a linear hyperbolic system of balance laws.

Fig

  Fig. 1. Time-evolution of functions V . Theorem 1 holds. The function ε used in the triggering condition (12) is chosen to be ε(t) = ςV (1)e -ηt , t ∈ R + with η = 1, V (1) = 0.6390 and ς is such that ςV (1) = 1 × 10 -2 . The number of events under this event-based approach was 109, counting them from t ≥ 1 λ = 1.

Fig. 2 .

 2 Fig. 2. Density of the inter-execution times with controller u = ϕ s (z) (left) and with controller u = ϕ d (z) (right).

This work has been partially supported by the LabEx PERSYVAL-Lab (ANR-11-LABX-0025-01) funded by the French program Investissements d'avenir.