Dr Mark Van Der Linden

Ralf R Reinert

Keywords: pneumococcal conjugate vaccines, Streptococcus pneumoniae, acute otitis media, Germany pneumococcal conjugate vaccines, Streptococcus pneumoniae, acute otitis media, Germany Approximately 1850

This retrospective analysis examined the pneumococcal serotype distribution of acute otitis media in Germany from 1995 to 2007. Data from the German National Reference Centre for Streptococci included 512 cases of pneumococcal otitis media in children and adults. Infections were mainly seen in children aged <5 years, who represented 67.0% of all reported cases. Most isolates (86.7%) were from spontaneous ruptures of the tympanum; 11.1% of the isolates were from otogenic sepsis or meningitis. Serotype 19F was the leading serotype (21.5%); serotype 3 (13.9%) was also often encountered. In children aged <5 years, the 7-valent, 10-valent, and 13-valent pneumococcal conjugate vaccines covered 54.3%, 60.2%, and 84.6% of the serotypes, respectively. Reduced penicillin susceptibility (minimum inhibitory concentration ≥0.1 mg/L) was seen in 11.0% of strains; 22.4% of strains were resistant to macrolides. Although based on a very limited selection of acute otitis media isolates, this analysis provides an estimate of the pneumococcal serotypes responsible for otitis media in Germany and underscores the need for future prospective studies.

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Reviewers' comments:

Reviewer #1: This is an important collection of AOM isolates. Such collections are rare since in most countries no central collection of AOM isolates exists. Therefore, despite several important limitations of the study, some of which are recognized by the authors and some of which I have added, this set of data is important. At least serotypes associated with complicated otitis media can be assumed from this collection. I. Abstract 1) Please provide the years of the study. We thank the reviewer for this comment. We have added the years of the study (1995)(1996)(1997)(1998)(1999)(2000)(2001)(2002)(2003)(2004)(2005)(2006)(2007) to the abstract.

2) You need to recognize already at the abstract level that this represents an important, but very selective series of isolates (since it is mainly or at least commonly non-responsive AOM, recurrent AOM, complicated AOM or invasive cases) We agree with the reviewers and have added the following sentence to the abstract: 'Although based on a very limited selection of acute otitis media isolates,....' II. Introduction 1) First page of the introduction lines, 14-15: There are new and more detailed references for the statement that serotypes are important in AOM (eg. The reviewer is right and we have added these more recent references to the text.

III. Results

1) The inclusion of S. pneumoniae isolated from blood and CSF is probably not appropriate since you do not have the proof that the otitis media was pneumococcal otitis media. I believe this should be excluded. Even though it was not microbiologically proven that AOM patients with pneumococci isolated from invasive material suffered from pneumococcal otitis, this is very likely. To emphasize this point we have added the following sentence to the manuscript: 'Even though it is highly likely that these children suffered from pneumococcal AOM, this was not microbiologically confirmed in all cases.' 2) Figure 1: Please instead of 2 categories "all ages" and "<5 years" modify those to "<5 years" and ">= 5 years". We have modified Table 2 according to the reviewers' comments. Furthermore we have changed the age cohorts accordingly ( 0 -<2, >=2 -<5, >=5 -<16, >=16) throughout the whole text and recalculated all corresponding values for coverage, serotypes, MICs, etc. as well as Figure 2.

3) Table 2 and text" Coverage by PCV7, PCV10 and PCV13: Some assumptions with regard to coverage of PCV7 and PCV10 against 6A must be made. All coverage data were calculated without assuming any cross-protection. A remark stating this explicitly concerning serotype 6A was added to the text: 'The coverage calculations were performed without accounting for possible cross-protection of serotype 6B towards serotype 6A, as described in some reports on PCV7 and the 10-valent vaccine'.

4) This study does not really represent all cases of AOM, but rather selected population with complicated AOM. Therefore, the findings may not necessarily compare to other studies, many of which also have their own selection. We fully agree with the reviewer regarding the limitations of this study, and that is why we have explicitly mentioned them in the discussion and also in the Abstract (see comment 1, 2)

Serotype distribution in pneumococcal acute otitis media with ruptured tympanic membrane or sepsis in Germany M. van der Linden, R. R. Reinert

Introduction

Streptococcus pneumoniae is a major pathogen in cases of acute otitis media (AOM), along with Haemophilus influenzae and Moraxella catarrhalis [START_REF] Fletcher | Brief review of the clinical effectiveness of PREVENAR ® against otitis media[END_REF]. Of these three organisms, S. pneumoniae has been associated with more severe AOM signs and symptoms, greater virulence, and more complications [START_REF] Palmu | Association of clinical signs and symptoms with bacterial findings in acute otitis media[END_REF]. To date, 91 distinct serotypes of S. pneumoniae have been identified, but only a few are responsible for most cases of AOM in young children, including serotypes 3, 6B, 9V, 14, 19F, and 23F [START_REF] Block | Pneumococcal serotypes from acute otitis media in rural Kentucky[END_REF][START_REF] Butler | Serotype distribution of Streptococcus pneumoniae infections among preschool children in the United States, 1978-1994: implications for development of a conjugate vaccine[END_REF].

The currently approved seven-valent pneumococcal conjugate vaccine (PCV7; Prevnar ® /Prevenar ® [Pfizer Inc, Philadelphia, PA]) includes purified capsular polysaccharides of seven serotypes of S. pneumoniae (4, 6B, 9V, 14, 18C, 19F, and 23F) conjugated to mutant diphtheria toxoid cross-reactive material (CRM) 197 . In prelicensure trials, PCV7 demonstrated almost 100% efficacy against invasive pneumococcal disease (IPD) and some efficacy against AOM [START_REF] Black | Efficacy, safety and immunogenicity of heptavalent pneumococcal conjugate vaccine in children. Northern California Kaiser Permanente Vaccine Study Center Group[END_REF]. In the Northern California Kaiser Permanente trial in the USA (n=37,868 infants), PCV7 was 7% efficacious in preventing episodes of otitis media [START_REF] Black | Efficacy, safety and immunogenicity of heptavalent pneumococcal conjugate vaccine in children. Northern California Kaiser Permanente Vaccine Study Center Group[END_REF]. In the Finnish Otitis Media (FinOM) trial (n=1,662 infants), efficacy against AOM from any cause was 6% [START_REF] Eskola | Efficacy of a pneumococcal conjugate vaccine against acute otitis media[END_REF]. Long-term follow-up of these cohorts confirmed that receipt of PCV7 in infancy was associated with fewer otitis media physician visits and episodes [START_REF] Fireman | Impact of the pneumococcal conjugate vaccine on otitis media[END_REF] and a reduction in tympanostomy tube placement in children aged <12 months to 4-5 years [START_REF] Fireman | Impact of the pneumococcal conjugate vaccine on otitis media[END_REF][START_REF] Palmu | The seven-valent pneumococcal conjugate vaccine reduces tympanostomy tube placement in children[END_REF].

In a more recent publication, Shouval et al. reported that in children aged <3 years in Israel, PCV7, 10-valent vaccine, and 13-valent vaccine provided coverage of AOM of 54%, 59%, and 79%, respectively [START_REF] Shouval | Serotype coverage of invasive and mucosal pneumococcal disease in Israeli children younger than 3 years by various pneumococcal conjugate vaccines[END_REF]. In a review by Rodgers et al., serotypes 6A, 6B, 14, 19A, 19F, and 23F were described as the most common serotypes in pneumococcal AOM among children worldwide [START_REF] Rodgers | Global serotype distribution among Streptococcus pneumoniae isolates causing otitis media in children: potential implications for pneumococcal conjugate vaccines[END_REF].

Although vaccine trials with PCV7 showed modest efficacy against AOM overall, there was considerable variation in serotype-specific efficacy. In the FinOM trial, PCV7 reduced the incidence of AOM due to vaccine serotypes by 57% overall. However, efficacy for specific serotypes ranged from 25% for serotype 19F to 84% for serotype 6B [START_REF] Eskola | Efficacy of a pneumococcal conjugate vaccine against acute otitis media[END_REF].

Following generalized vaccination with PCV7 in the USA, at least three studies reported on the impact of immunization on the microbiology of AOM. These studies analyzed pre-and post-vaccination rates of middle-ear fluid (MEF) pathogen isolates and found that community use of PCV7 significantly reduced the overall proportion of S. pneumoniae isolates and vaccine serotypes causing AOM [START_REF] Block | Community-wide vaccination with the heptavalent pneumococcal conjugate significantly alters the microbiology of acute otitis media[END_REF][START_REF] Casey | Changes in frequency and pathogens causing acute otitis media in 1995-2003[END_REF][START_REF] Mcellistrem | Epidemiology of acute otitis media caused by Streptococcus pneumoniae before and after licensure of the 7-valent pneumococcal protein conjugate vaccine[END_REF]. Data from the USA demonstrated that in a comparison of 2004 and 1997-1999 (baseline period), rates of ambulatory visits and antibiotic prescriptions attributable to AOM decreased by 42.7% and 41.9%, respectively [START_REF] Zhou | Trends in acute otitis media-related health care utilization by privately insured young children in the United States, 1997-2004[END_REF]. Notably, the proportion of AOM infections due to nonvaccine pneumococcal strains and serogroups 6 and 19 increased over time [START_REF] Block | Community-wide vaccination with the heptavalent pneumococcal conjugate significantly alters the microbiology of acute otitis media[END_REF][START_REF] Mcellistrem | Epidemiology of acute otitis media caused by Streptococcus pneumoniae before and after licensure of the 7-valent pneumococcal protein conjugate vaccine[END_REF].

In July 2006, the German Ständige Impfkomission (permanent committee on vaccination, STIKO) recommended vaccination with PCV7 for all children aged <2 years [START_REF] Claes | Health Technology Assessment Heptavalenter Pneumokokkenkonjugat-Impfstoff (PCV7)[END_REF]. However, little is known about the epidemiology of pneumococcal otitis media in Germany [START_REF] Rodgers | Global serotype distribution among Streptococcus pneumoniae isolates causing otitis media in children: potential implications for pneumococcal conjugate vaccines[END_REF][START_REF] Ziebold | Epidemiology of pneumococcal disease in children in Germany[END_REF], and no recent data are available on the serotype distribution of pneumococcal isolates from German children and adults with otitis media. Therefore, our study examined the pneumococcal serotype distribution related to AOM, using data from the German National Reference Centre for Streptococci.

Materials and methods

The database of the German National Reference Centre for Streptococci, which contains information on more than 25,000 cases of pneumococcal infections, was searched for cases of pneumococcal otitis media that occurred between 1995 and 2007. All S. pneumoniae isolates from ear swab and other isolates with the clinical diagnosis -acute otitis‖ or -mastoiditis‖ were included in the study. Duplicate isolates were excluded from the analysis.

Strains were obtained from (1) the study of IPD in children performed on a nationwide basis in Germany since 1996 (collaboration between the Robert Koch Institute, the ESPED [Erhebungseinheit für Seltene Pädiatrische Erkrankungen in Deutschland], and the National Reference Centre for Streptococci) [START_REF] Reinert | Antimicrobial resistance of Streptococcus pneumoniae recovered from outpatients with respiratory tract infections in Germany from 1998 to 1999: results of a national surveillance study[END_REF][START_REF] Reinert | Molecular epidemiology of penicillin-nonsusceptible Streptococcus pneumoniae isolates from children with invasive pneumococcal disease in Germany[END_REF][START_REF] Siedler | Regional differences in the epidemiology of invasive pneumococcal disease in toddlers in Germany[END_REF][START_REF] Von Kries | Will the 7valent pneumococcal vaccine have a similar impact on all invasive pneumococcal infections in children in Germany as in the Kaiser Permanente Trial?[END_REF][START_REF] Von Kries | Prediction of the potential benefit of different pneumococcal conjugate vaccines on invasive pneumococcal disease in German children[END_REF][START_REF] Von Kries | Proportion of invasive pneumococcal infections in German children preventable by pneumococcal conjugate vaccines[END_REF], and (2) the nation-wide surveillance of IPDs in adults [START_REF] Reinert | Invasive pneumococcal disease in adults in North-Rhine Westphalia, Germany, 2001-2003[END_REF] and upper respiratory tract infections in children and adults [START_REF] Reinert | Antimicrobial resistance of Streptococcus pneumoniae recovered from outpatients with respiratory tract infections in Germany from 1998 to 1999: results of a national surveillance study[END_REF]. All studies were summarized in one database and made available for analysis.

Pneumococcal isolates were identified with standard microbiology methods, such as determination of optochin susceptibility and bile solubility. Strains were typed using the antisera provided by the Statens Serum Institut in Copenhagen, Denmark. A subgroup of all strains was tested for antibiotic sensitivity to penicillin G and clarithromycin (macrolide) using the Clinical and Laboratory Standards Institute (CLSI) guidelines for susceptibility testing [START_REF]Performance standards for antimicrobial disk susceptibility tests; approved standard-ninth edition[END_REF].

Results

A total of 512 isolates, mainly from children aged <2 years (34.4%), were included in the study. The remainder were from children aged ≥2 to <5 years (32.6%); ≥5 to <16 years (22.5%); and adults aged ≥16 years (10.5%). Sources of pneumococcal isolates are presented in Table 1. Most cases (444 strains; 86.7% of isolates) were from children with spontaneous rupture of the tympanum (specimen = pus from the middle ear or ear swab). In addition, three isolates from puncture of the mastoid and eight isolates from tympanocentesis were included in the study. There were 57 isolates (11.1%) from invasive disease (sepsis and meningitis).

Even though it is highly likely that these children suffered from pneumococcal AOM, this was not microbiologically confirmed in all cases. Infections were more often seen in males (54.5% of strains).

Serotype data for all strains are presented in Figure 1. In our study, 19F was the leading serotype (21.5%), followed by serotypes 3 (13.9%), 14 (10.5%), and 23F (10.0%).

Among children aged <5 years, the leading serotype was 19F (24.5%), followed by serotypes 14 (12.2%), 23F (11.4%), and 3 (8.7%) (Figure 1).

Coverage by serotypes included in the current and investigational pneumococcal conjugate vaccine formulations was evaluated. These formulations include PCV7 (serotypes 4, 6B, 9V, 14, 18C, 19F, and 23F); the 10-valent vaccine (PCV7 serotypes plus serotypes 1, 5, and 7F); and the investigational 13-valent vaccine (10-valent vaccine serotypes plus serotypes 3, 6A, and 19A). Coverages of the 7-valent, 10-valent, and investigational 13valent pneumococcal conjugate vaccines in isolates from pneumococcal AOM with ruptured tympanic membrane or sepsis in Germany are presented in Table 2. The coverage calculations were performed without accounting for possible cross-protection of serotype 6B towards serotype 6A, as described in some reports on PCV7 and the 10-valent vaccine [START_REF] Whitney | Effectiveness of seven-valent pneumoccoccal conjugate vaccine against invasive pneumococcal disease: a matched case-control study[END_REF][START_REF] Vesikari | Immunogenicity of the 10-valent pneumococcal non-typeable Haemophilus influenzae protein D conjugate vaccine (PHiD-CV) compared to the licensed 7vCRM vaccine[END_REF]. For children aged <5 years, coverage of PCV7 was 60.3% (n=207); the 10-valent vaccine, 63.6% (n=218); and the 13-valent vaccine, 85.4% (n=293). The cumulative coverage of the different pneumococcal conjugate vaccine formulations is presented in Figure 2.

Antibiotic susceptibility data were available for 456 isolates. Reduced susceptibility to penicillin was seen in 11.0% of strains. This included a relatively high rate of strains (4.4%) with high-level penicillin G resistance (minimum inhibitory concentration [MIC] ≥2 mg/L).

The rate of resistance to macrolides (clarithromycin) was 22.4% (Table 3). Antibiotic susceptibility data were available for 304 strains isolated from children aged <5 years; 9.5% and 24.0% of these strains were found to be nonsusceptible to penicillin G and clarithromycin, respectively. Characteristics of IPD in 33 children with otitis media (otogenic sepsis or meningitis) are presented in Table 4. These cases were mainly seen in very young children (aged ≤1 year). The collection included 13 cases diagnosed from cerebrospinal fluid and 20 cases from blood. Among these infections, serotype 14 was the most common serotype (14 cases) followed by serotype 19A (4 cases). Serotype 19A is not included in PCV7 or the 10-valent pneumococcal conjugate vaccine. Of note, macrolide resistance was relatively common in those cases and occurred in more than a third of isolates.

Discussion

This study presents data on the serotype distribution of pneumococcal isolates from children and adults with spontaneously draining otitis media in Germany. The analysis included 512 cases, with 67.0% occurring in children aged <5 years. The results showed a relatively high rate of serotype 19F infections (21.5%), which is in accordance with the serotype distribution found in recent clinical studies and postlicensure surveillance of AOM in children in Finland [START_REF] Eskola | Efficacy of a pneumococcal conjugate vaccine against acute otitis media[END_REF], the Czech-Republic [START_REF] Prymula | Pneumococcal capsular polysaccharides conjugated to protein D for prevention of acute otitis media caused by both Streptococcus pneumoniae and non-typable Haemophilus influenzae: a randomised double-blind efficacy study[END_REF], and the USA [START_REF] Hausdorff | Multinational study of pneumococcal serotypes causing acute otitis media in children[END_REF]. However, some differences in serotype distribution, particularly regarding serotype 23F infections, were noted in the German data compared with distribution data from other countries.

A 2002 retrospective study, for example, analyzed nine geographically diverse data sets comprising pneumococcal isolates from MEF samples collected from children with AOM [START_REF] Hausdorff | Multinational study of pneumococcal serotypes causing acute otitis media in children[END_REF]. Samples were collected between 1994 and 2000 from 3,232 children in the USA, Argentina, Finland, France, Greece, Israel, and several Eastern European countries. The results showed that the most common serotypes in each data set and age group tended to be the same, despite variations in location, data-set design, and antibiotic resistance. Serotypes 23F, 19F, and 14 were most prominent, each comprising about 13% to 16% of the overall data set [START_REF] Hausdorff | Multinational study of pneumococcal serotypes causing acute otitis media in children[END_REF]. Serotypes 6B, 6A, 19A, and 9V were less common, each comprising about 5% to 10% of all data sets. Serotypes 1, 5, and 7 contributed little (<2%) to overall pneumococcal AOM. Serotype 3 represented about 4% of the overall serotypes, and was most prominent in the youngest (<6 months) and oldest (≥60 months) age groups (approximately 5.8% and 21.5% of all isolates, respectively) [START_REF] Hausdorff | Multinational study of pneumococcal serotypes causing acute otitis media in children[END_REF].

In contrast to these observations, our analysis of data from German children and adults showed a lower rate of serotype 23F infections (10.0%) and a higher rate of serotype 3 infections (13.9%). Globally, serotype 3 has become an increasingly significant MEF pathogen in children [START_REF] Rodgers | Global serotype distribution among Streptococcus pneumoniae isolates causing otitis media in children: potential implications for pneumococcal conjugate vaccines[END_REF]. In German children aged <5 years, serotype 3 was found in 8.7% of cases. The coverage for the serotypes included in PCV7 (recommended for general use in children aged <2 years in Germany) was 60.3%, which was lower than the serotype coverage previously reported for IPD in Germany [START_REF] Reinert | Molecular epidemiology of penicillin-nonsusceptible Streptococcus pneumoniae isolates from children with invasive pneumococcal disease in Germany[END_REF]. The inclusion of serotypes 1, 5, and 7F, which are in the 10-valent pneumococcal conjugate vaccine, increases the coverage by 3.3% to 63.6%. The investigational 13-valent vaccine, which includes serotypes 3, 6A, and 19A in addition to the 10-valent vaccine serotypes, increases the coverage to 85.4%, mainly due to the inclusion of serotypes 3 and 19A (Figure 2). Interestingly, we found a relatively high rate of reduced penicillin susceptibility (11.0%) in the present study. This rate was higher than that previously reported with invasive disease in German children [START_REF] Reinert | Molecular epidemiology of penicillin-nonsusceptible Streptococcus pneumoniae isolates from children with invasive pneumococcal disease in Germany[END_REF]. In addition, we found a high rate of macrolide resistance (22.4%), in accordance with rates reported for IPD in children in Germany [START_REF] Reinert | Molecular epidemiology of penicillin-nonsusceptible Streptococcus pneumoniae isolates from children with invasive pneumococcal disease in Germany[END_REF][START_REF] Siedler | Regional differences in the epidemiology of invasive pneumococcal disease in toddlers in Germany[END_REF]. These high resistance rates may reflect the complexity of many of the cases, which included otitis media therapy failures.

Our study has some limitations. First, it covers a long period of time, and the power is low. Admittedly, more patients and data would have been useful for our analysisunfortunately we had no control over the number of the study population and were forced to rely only on the AOM serotype data available for Germany at the time of the study. A second limitation is the fact that microbiological diagnosis is not done on a routine basis in Germany.

Consequently, the incidence of AOM in Germany is probably under-reported, because only a portion of the thousands of children who contract AOM in Germany each year are tested.

Mild IPD may be under-diagnosed in several European countries, because blood cultures are not usually ordered for outpatients, who may be treated with no documentation of their febrile episodes. Blood cultures may not be ordered for inpatients with fever without focus or with pneumonia [START_REF] Vergison | Streptococcus pneumoniae incidence in Western Europe[END_REF]. Lastly, pneumococcal isolates used in our study were identified with standard microbiology methods. Use of more specific techniques, such as polymerase chain reaction (PCR), would have improved identification of otitis media-associated bacteria. Despite these limitations, our retrospective analysis of pneumococcal strains gathered the data needed to provide a rough estimate of the pneumococcal serotypes that cause otitis media associated with spontaneous drainage in Germany. This information will be invaluable when planning vaccination strategies to protect against otitis media.

  Ref.: Ms. No. EJCMID-D-09-00590 Serotype distribution in pneumococcal acute otitis media with ruptured tympanic membrane or sepsis in Germany European Journal of Clinical Microbiology & Infectious Diseases Dear Dr van der Linden,

Fig. 1 Fig. 2

 12 Fig. 1 Serotype Distribution for Pneumococcal OM Germany All Ages and <5 years

  Figure 1 Click here to download high resolution image

  Shouval et al, PIDJ 2009;28:277-282; Aguilar et al, J Pediatr Otolaryngol 73:1407-1411, 2009; Rodgers et al, Vaccine 27:3802-10, 2009).

Table 1

 1 Sources of pneumococcal isolates from children with pneumococcal otitis media in

	Germany
	Specimen	Number (%)
	Blood	24 (4.7)
	Cerebrospinal fluid	33 (6.4)
	Ear (tympanocentesis)	8 (1.6)
	Mastoid	3 (0.6)
	Pus middle ear	13 (2.5)
	Swab ear	431 (84.2)
	Total	512 (100)

Table 2

 2 Coverage of the 7-valent, 10-valent, and investigational 13-valent pneumococcal conjugate vaccines in pneumococcal acute otitis media with ruptured tympanic membrane or sepsis isolates in Germany

	Age Group	Vaccine formulation 7-valent 10-valent 13-valent	Total
	<5 years (n, %)	207 (60.3) 218 (63.6) 293 (85.4) 343 (100)
	≥5 years (n, %)	71 (42.0)	90 (53.3) 140 (82.8) 169 (100)
	All ages (n, %)	278 (54.3) 308 (60.2) 433 (84.6) 512 (100)

Table 3

 3 Antibiotic resistance in S. pneumoniae isolated from patients with acute otitis media

	in Germany (n=465)
		Penicillin G	Clarithromycin
	Resistance	n (%)	n (%)
	Susceptible	406 (89.0)	349 (76.5)
	Intermediate	30 (6.6)	5 (1.1)
	Resistant	20 (4.4)	102 (22.4)
	Total	456 (100)	456 (100)

Table 4

 4 Characteristics of invasive pneumococcal disease in children with pneumococcal otitis media (otogenic sepsis or meningitis) in Germany (n=33)

		Patients (n=33)		Susceptibility
	Age, Gender	Diagnosis (Isolate)	Serotype	E-myc	Pen G
	5 mo, ND	Otitis perforans (B)	33A	S	S
	6 mo, M	Mastoiditis (B)	23F	S	S
	2 y, F	Meningitis, mastoiditis (CSF)	24F	S	S
	9 mo, F	ND (B)	19A	R	S
	3 y, M	Pneumonia, convulsion (B)	19A	R	S
	4 mo, M	Meningitis (CSF)	19A	ND	ND
	9 mo, F	Meningitis, mastoiditis (CSF)	19A	S	S
	1 y, F	ND (B)	19F	S	S
	3 y, M	ND (B)	19F	S	I
	1 y, M	Sepsis (B)	18C	S	S
	2 y, M	Meningitis (CSF)	18C	S	S
	1 y, F	ND (B)	14	R	S
	2 y, F	Pharyngitis (B)	14	R	S
	2 y, F	Pharyngitis (B)	14	R	S
	1 y, F	Pneumonia, coxitis (B)	14	R	S
	2 y, F	High fever (B)	14	R	S
	1 y, F	Sepsis (B)	14	R	S
	1 y, M	Mastoiditis (B)	14	R	S
	1 y, F	Meningitis (B)	14	R	S
	1 y, F	ND (B)	14	S	S
	1 y, F	Meningitis (CSF)	14	S	S
	1 y, M	Meningitis (CSF)	14	S	R
	7 mo, M	ND (CSF)	14	S	S
	1 y, M	Meningitis (CSF)	14	R	S
	1 y, M	Meningitis (CSF)	14	R	S
	1 y, M	Meningitis (CSF)	10A	S	S
	11 mo, M	Recurrent otitis (B)	9N	ND	ND
	4 y, ND	Chronic otitis (B)	7F	S	S
	11 mo, M	Meningitis (CSF)	7F	S	S
	1 y, M	Meningitis (CSF)	6A	S	S
	2 y, M	Sepsis, meningitis (B)	6B	S	S
	3 y, F	ND (B)	6B	S	S
	2 y, F	Meningitis, sepsis (CSF)	4	S	S

E-myc erythromycin, Pen G penicillin G, B blood, CSF cerebrospinal fluid, S susceptible, R resistant, I intermediate, ND no data

Acknowledgements Medical writing support for this manuscript was provided by Prasad Kulkarni of Excerpta Medica (Bridgewater, NJ), and was funded by Wyeth, which was acquired by Pfizer Inc in October 2009.

Conflicts of Interest Dr

Reinert is an employee of Pfizer Inc. Dr van der Linden has been a member of advisory boards for Wyeth and GlaxoSmithKline, and has received invitations to scientific conferences from Wyeth and GlaxoSmithKline.


Martin Gugat

Mario Sigalotti

STARS OF VIBRATING STRINGS: SWITCHING BOUNDARY FEEDBACK STABILIZATION

Keywords: 2000 Mathematics Subject Classification. Primary: 49K20, 35L05 hyperbolic pde, network, feedback stabilization of pdes, switching control, wave equation, switching feedback, robustness

Stars of vibrating strings: switching boundary feedback stabilization

Martin Gugat, Mario Sigalotti

1. Introduction. We consider a star-shaped network of N ≥ 3 finite strings (with possibly different lengths) that are governed by the wave equation. At the boundary point zero the strings are coupled. At the other end of each string a feedback law is prescribed that requires the time derivative at this point to be proportional to the space derivative at this point. For a single string, this feedback law has been considered in [START_REF] Cox | The rate at which energy decays in a string damped at one end[END_REF], and it has been shown that the energy vanishes in finite time. In [START_REF] Gugat | Optimal boundary feedback stabilization of a string with moving boundary[END_REF] it is shown that the result from [START_REF] Cox | The rate at which energy decays in a string damped at one end[END_REF] is stable in the sense that also with moving boundaries, the energy is driven to zero in finite time. In this paper we show that also on the network, the energy is driven to zero in finite time if the feedback control is active on all N boundary nodes for a sufficiently long time. Our particular interest in this paper is the question: What happens if at one of the nodes the feedback control becomes inactive? This need not be a fixed boundary node on the whole time interval but the inactivity may switch between different boundary nodes in time. The idea is that at each moment, it may happen that one of the N controllers is inactive, and we still want to have a stable system.

The boundary control of the wave equation has been studied by many authors (see e.g. [START_REF] Russell | Nonharmonic Fourier series in the control theory of distributed parameter systems[END_REF], [START_REF] Lions | Exact controllability, stabilization and perturbations for distributed systems[END_REF], [START_REF] Krabs | Optimal control of processes governed by partial differential equations. II. Vibrations[END_REF], [START_REF] Krabs | On moment theory and controllability of one-dimensional vibrating systems and heating processes[END_REF], [START_REF] Avdonin | Families of exponentials[END_REF], [START_REF] Zuazua | Optimal and approximate control of finite-difference approximation schemes for the 1D wave equation[END_REF] and the references therein). A problem of optimal switching boundary control of a single string to rest in finite time has been considered in [START_REF] Gugat | Optimal switching boundary control of a string to rest in finite time[END_REF], where a string with boundary control at both ends has been considered and, at each moment, at most one of the controls is allowed to be active. The corresponding problem for the heat equation has been analyzed in [START_REF] Zuazua | Switching controls[END_REF] using an adapted adjoint calculus.

Networks of strings have been considered for example in [START_REF] Leugering | On feedback controls for dynamic networks of strings and beams and their numerical simulation[END_REF][START_REF] Leugering | On exact controllability of generic trees, in "Contrôle des systèmes gouvernés par des équations aux dérivées partielles (Nancy[END_REF][START_REF] Ammari | Stabilization of generic trees of strings[END_REF][START_REF] Lagnese | Modeling, analysis and control of dynamic elastic multi-link structures[END_REF][START_REF] Valein | Stabilization of the wave equation on 1-d networks[END_REF]] and an overview is given in [START_REF] Dáger | Wave propagation, observation and control in 1-d flexible multistructures[END_REF]. In these works about networks, the nodes where a feedback control acts on the system are constant during the control process.

In contrast to this situation, in this paper we consider a system where these nodes may change as time proceeds. We are interested in the question: How many feedback controls must be switched on at each moment in time to achieve exponential decay? We show that N -1 controls are sufficient. It is essential that the choice of the inactive control need not be constant but can vary in a quite general way with time.

In a similar spirit in [START_REF] Haraux | Asymptotic stability for intermittently controlled second-order evolution equations[END_REF] it has been studied how often the control should be active in order to stabilize a string with interior damping. Analogous questions have been addressed in [START_REF] Chaillet | Uniform stabilization for linear systems with persistency of excitation: the neutrally stable and the double integrator cases[END_REF][START_REF] Chitour | On the stabilization of persistently excited linear systems[END_REF] for finite dimensional systems.

This paper has the following structure: First we define the problem of switching feedback boundary stabilization of a network of strings. Then we state our main results, which are two sufficient conditions for exponential decay of the derivatives in our system. First we state a backwards in time condition and then we state a forward condition.

The proofs of the sufficiency of both conditions take advantage of the reformulation of the initial boundary value problem in terms of Riemann invariants. We show that if N feedback nodes are active for a sufficiently long time interval, after finite time the partial derivatives of the solution are equal to zero. For the backward condition, we show that if at each moment in time only one wave arrives at the coupling note of the network, the partial derivatives of the solution go to zero exponentially fast. The proof of the sufficiency of the forward condition is based upon the construction of a suitable Lyapunov function.

2. The System. Let N ≥ 3 and consider N strings of length L i > 0 (i ∈ {1, ..., N }). Define L = max{L 1 , ..., L N }. Let the corresponding wave speeds c i > 0 be given. Define c = min{c 1 , ..., c N }. For i ∈ {1, ..., N } define the sets

Ω i = (0, ∞) × (0, L i ). Define the set B = {(y (i) 0 , y (i) 1 ) N i=1 : ∂ x y (i) 0 ∈ L ∞ (0, L i ), y (i) 1 ∈ L ∞ (0, L i ), i ∈ {1, ..., N }, y (i) 0 (0) = y (j) 0 (0), i, j ∈ {1, ..., N }}.
For i ∈ {1, ..., N }, let σ i : (0, ∞) → {0, 1} be a measurable function. The equation σ i (t) = 0 will indicate that at time t the feedback at the end of string i is not active, whereas σ i (t) = 1 means that the feedback is active.

Let feedback parameters κ i > 0 be given. For (y

(i) 0 , y (i)
1 ) N i=1 ∈ B we consider the system (S) given by the equations

v (i) (0, x) = y (i) 0 (x), v (i) t (0, x) = y (i) 1 (x), x ∈ (0, L i ), i ∈ {1, ..., N } (1)
v (i) tt (t, x) = c 2 i v (i) xx (t, x), (t, x) ∈ Ω i , i ∈ {1, ..., N } (2)
v (i) (t, 0) = v (j) (t, 0), t ∈ (0, ∞), i, j ∈ {1, ..., N } (3)
c 1 v (1) x (t, 0) + c 2 v (2) x (t, 0) + • • • + c N v (N ) x (t, 0) = 0, t ∈ (0, ∞) (4)
c i v (i) x (t, L i ) = -σ i (t) κ i v (i) t (t, L i ), t ∈ (0, ∞), i ∈ {1, ..., N }. (5)
Conditions (1) describe the initial state of the system. The dynamics on the strings is given by the wave equation [START_REF] Avdonin | Families of exponentials[END_REF]. Equations ( 3) and (4) describe how the strings are coupled. The feedback control (that is switched off if σ i (t) = 0) is given by [START_REF] Cox | The rate at which energy decays in a string damped at one end[END_REF].

Remark 1. With homogeneous Neumann conditions (that is σ i (t) ≡ 0) at all nodes, the coupling conditions (3) and ( 4) guarantee the conservation of the energy of the system (see Section 2.2 for details).

2.1. Solution of the System. In this section we define the d'Alembert solution in the sense of characteristics for system (S) analogously to Theorem 1 in [START_REF] Gugat | Boundary Feedback Stabilization by Time Delay for One-Dimensional Wave Equations[END_REF].

Define the number λ min = min{L 1 /c 1 , ..., L n /c n }.

For i ∈ {1, ..., N } let Ω i = [0, ∞) × [0, L i ]. Define the orthogonal symmetric N × N reverberation matrix A = N -2 N        1 2 2-N 2 2-N . . . 2 2-N 2 2-N 1 2 2-N . . . 2 2-N . . . . . . . . . 2 2-N . . . 2 2-N 1 2 2-N 2 2-N 2 2-N . . . 2 2-N 1        . (6)
Theorem 2.1. [Well-posedness of (S)] Let the initial state (y

(i) 0 , y (i)
1 ) N i=1 ∈ B be given. For x ∈ [0, L i ] define the functions α i , β i by

α i (x) = 1 2 y (i) 0 (x) + 1 2c i x 0 y (i) 1 (s) ds, x ∈ (0, L i ), (7)
β i (x) = 1 2 y (i) 0 (x) - 1 2c i x 0 y (i) 1 (s) ds, x ∈ (0, L i ). ( 8
)
For t ∈ [L i /c i , 2L i /c i ] let α ′ i (c i t) =   κ i σ i t -Li ci - 1
κ i σ i t -Li ci + 1   β ′ i (2L i -c i t) . ( 9
)
Together with [START_REF] Gugat | Optimal boundary feedback stabilization of a string with moving boundary[END_REF] this yields the values of

α ′ i (c i t) for t ∈ [0, 2λ min ]. For t ∈ [0, 2λ min ] let the equation      c 1 β ′ 1 (-c 1 t) c 2 β ′ 2 (-c 2 t) . . . c N β ′ N (-c N t)      = A      c 1 α ′ 1 (c 1 t) c 2 α ′ 2 (c 2 t) . . . c N α ′ N (c N t)      (10)
define the values of 9) yields the values of α ′ i (c i t). Hence for t ∈ [0, 4λ min ] the values of α ′ i (c i t) are well-defined. Now [START_REF] Haraux | Asymptotic stability for intermittently controlled second-order evolution equations[END_REF] defines the values of β ′ i (c i t) for t ∈ [-4λ min , 0). 9) yields the values of α ′ i (c i t). Hence for t ∈ [0, 6λ min ] the values of α ′ i (c i t) are well-defined. Now [START_REF] Haraux | Asymptotic stability for intermittently controlled second-order evolution equations[END_REF] defines the values of β ′ i (c i t) for t ∈ [-6λ min , 0). By repeating the process we define the functions α ′ i (c i t), β ′ i (-c i t) inductively and obtain functions

β ′ i (c i t) for t ∈ [-2λ min , 0). For t ∈ [2L i /c i , 2λ min + 2L i /c i ] equation (
For t ∈ [2L i /c i + 2λ min , 2L i /c i + 4λ min ] equation (
α ′ i ∈ L ∞ loc (0, ∞) and β ′ i ∈ L ∞ loc (-∞, L i ). For x > L i define α i (x) = α i (L i ) + x Li α ′ i (s) ds. For x < 0, let β i (x) = β i (0) + x 0 β ′ i (s) ds. Then (v (i) ) N i=1 given by v (i) (t, x) = α i (x + c i t) + β i (x -c i t) (11)
are solutions of (S) in the sense described below. The function v (i) is continuous on Ω i and v

(i) t , v (i) x ∈ L ∞ loc (Ω i ).
Given the family of test functions

T i = {ϕ ∈ C 2 (Ω i ) : There exists a set Q = [t 1 , t 2 ] × [x 1 , x 2 ] ⊂ Ω i
such that the support of ϕ is contained in the interior of Q}, the function v (i) satisfies the wave equation ( 2) in the following weak sense:

Ωi v (i) t (t, x)ϕ t (t, x) d(t, x) = c 2 i Ωi v (i) x (t, x)ϕ x (t, x) d(t, x) for all ϕ ∈ T i .
The functions v (i) satisfy (1) for almost every x, (3) for every t, and ( 4), [START_REF] Cox | The rate at which energy decays in a string damped at one end[END_REF] for almost every t.

Proof. Due to [START_REF] Gugat | Optimal boundary feedback stabilization of a string with moving boundary[END_REF], [START_REF] Gugat | Optimal switching boundary control of a string to rest in finite time[END_REF], it is easy to check that (v (i) ) N i=1 satisfies the initial conditions (1). Similar as in the proof of Theorem 1 in [START_REF] Gugat | Boundary Feedback Stabilization by Time Delay for One-Dimensional Wave Equations[END_REF], integration by parts shows that v (i) satisfies the wave equation in the weak sense given in Theorem 2.1. Writing the boundary conditions (5) in terms of α ′ i and β ′ i yields equation [START_REF] Gugat | Boundary Feedback Stabilization by Time Delay for One-Dimensional Wave Equations[END_REF]. By definition of β ′ i (•), for almost every t > 0 we have [START_REF] Haraux | Asymptotic stability for intermittently controlled second-order evolution equations[END_REF]. Hence we have

      v (1) t (t, 0) v (2) t (t, 0) . . . v (N ) t (t, 0)       =      c 1 (α ′ 1 (c 1 t) -β ′ 1 (-c 1 t)) c 2 (α ′ 2 (c 2 t) -β ′ 2 (-c 2 t)) . . . c N (α ′ N (c N t) -β ′ N (-c N t))      = (I -A)      c 1 α ′ 1 (c 1 t) c 2 α ′ 2 (c 2 t) . . . c N α ′ N (c N t)      = N -2 N    2 N -2 2 N -2 . . . 2 N -2 . . . . . . . . . 2 N -2 2 N -2 . . . 2 N -2         c 1 α ′ 1 (c 1 t) c 2 α ′ 2 (c 2 t) . . . c N α ′ N (c N t)      , which implies the equation v (i) t (t, 0) = v (j)
t (t, 0) for t ∈ (0, ∞) and i, j ∈ {1, ..., N }. Due to the definition of the set B, this implies that (3) is valid. Moreover, we have

      c 1 v (1) x (t, 0) c 2 v (2)
x (t, 0) . . .

c N v (N ) x (t, 0)       =      c 1 (α ′ 1 (c 1 t) + β ′ 1 (-c 1 t)) c 2 (α ′ 2 (c 2 t) + β ′ 2 (-c 2 t)) . . . c N (α ′ N (c N t) + β ′ N (-c N t))      = (I + A)      c 1 α ′ 1 (c 1 t) c 2 α ′ 2 (c 2 t) . . . c N α ′ N (c N t)      = N -2 N       2N -2 N -2 2 2-N . . . 2 2-N 2 2-N . . . . . . . . . . . . 2 2-N 2 2-N . . . 2 2-N 2N -2 N -2            c 1 α ′ 1 (c 1 t) c 2 α ′ 2 (c 2 t) . . . c N α ′ N (c N t)      , which implies the equation c 1 v (1)
x (t, 0) + c 2 v

(2)

x (t, 0) + • • • + c N v (N )
x (t, 0) = 0 for almost t ∈ (0, ∞), hence (4) holds.

2.2. The Energy of the System. Define the energy of the solution of system (S) as given in Theorem 2.1 by

E(t) = 1 2 N i=1 c i Li 0 v (i) t (t, x) 2 c 2 i + v (i) x (t, x) 2 dx.
Equation [START_REF] Krabs | Optimal control of processes governed by partial differential equations. II. Vibrations[END_REF] implies

v (i) x (t, x) = α ′ i (x + c i t) + β ′ i (x -c i t), v (i) t (t, x) = c i [α ′ i (x + c i t) -β ′ i (x -c i t)]. (12) Hence we have v (i) t (t, x) 2 c 2 i + v (i) x (t, x) 2 = (α ′ i (x + c i t) + β ′ i (x -c i t)) 2 + (α ′ i (x + c i t) -β ′ i (x -c i t)) 2 = 2[α ′ i (x + c i t) 2 + β ′ i (x -c i t) 2
]. Thus we have

E(t) = N i=1 c i Li 0 α ′ i (c i t + x) 2 + β ′ i (-c i t + x) 2 dx = N i=1 c i Li+cit cit α ′ i (s) 2 ds + Li-cit -cit β ′ i (s) 2 ds
Hence, for almost every t ≥ 0 the time-derivative of the energy E(t) exists and is given by

Ė(t) = N i=0 c 2 i α ′ i (L i + c i t) 2 -β ′ i (L i -c i t) 2 -c 2 i α ′ i (c i t) 2 -β ′ i (-c i t) 2 = - N i=0 4κ i σ i (t) (κ i σ i (t) + 1) 2 c 2 i β ′ i (L i -c i t) 2 -(c 1 α ′ 1 (c 1 t), . . . , c N α ′ N (c N t)) 2 + (c 1 β ′ 1 (-c 1 t), . . . , c N β ′ N (-c N t)) 2 = - N i=0 4κ i σ i (t) (κ i σ i (t) + 1) 2 c 2 i β ′ i (L i -c i t) 2 , ( 13
)
where the last equality follows from [START_REF] Haraux | Asymptotic stability for intermittently controlled second-order evolution equations[END_REF] and the orthogonality of the matrix A. In particular, E is non-increasing. Notice that, in terms of v (i) ,

Ė(t) = - N i=0 κ i σ i (t) (κ i σ i (t) + 1) 2 c i v (i) x (t, L i ) -v (i) t (t, L i ) 2
for almost every t.

3. Main Results. In this section we state the main results of the paper, which provide conditions on the switching functions σ i that guarantee exponential decay of the first order derivatives of the solutions of system (S).

Theorem 3.1. [Switching feedback stabilization of (S): Backward Condition] Consider system (S) defined in ( 1)- [START_REF] Cox | The rate at which energy decays in a string damped at one end[END_REF]. Let

λ = max{L 1 /c 1 , . . . , L N /c N }.
If κ i = 1 for all i ∈ {1, ..., N } and there exists a time T > 0 such that

N i=1 σ i t - L i c i = N almost everywhere on (T, T + 2λ) ( 14
)
then the system reaches a constant state after finite time, in the sense that for all t > T + 2λ we have v

(i) x (t, x) = 0 and v (i) t (t, x) = 0, x ∈ (0, L i ), i ∈ {1, ..., N }. Define f = max{ 2 N , N -2 N }. If for all i ∈ {1, ..., N } we have κ i -1 κ i + 1 < 1 -f √ N and N i=1 σ i t - L i c i ≥ N -1 almost everywhere on (λ, ∞) (15)
then the system state converges exponentially fast to a constant state, in the sense that for almost every t in (0, ∞) we have the inequality

ess sup{|v (i) x (t, x)|, |v (i) t (t, x)| : x ∈ (0, L i ), i ∈ {1, ..., N }} ≤ C exp ln(F ) 2λ t ( 16
)
where

F = √ N max i∈{1,...,N } κi-1 κi+1 + f < 1.
The decay is uniform with respect to σ, that is the constant C in ( 16) is independent of the choice of σ verifying [START_REF] Leugering | On exact controllability of generic trees, in "Contrôle des systèmes gouvernés par des équations aux dérivées partielles (Nancy[END_REF].

Proof. The proof of the first assertion is given in Section 4.1. The proof of the second assertion is given in Section 4.2.

Theorem 3.2. [Switching feedback stabilization of (S): Forward Condition] Consider system (S) defined in (1)- [START_REF] Cox | The rate at which energy decays in a string damped at one end[END_REF]. If κ i = 1 for all i ∈ {1, ..., N } and there exists a time T > 0 such that

N i=1 σ i t + L i c i = N almost everywhere on (T, T + 2λ),
then the system reaches a constant state after finite time.

If κ i > 0 for all i ∈ {1, ..., N } and

N i=1 σ i t + L i c i ≥ N -1 almost everywhere on (0, ∞) (17)
then the energy of the state converges exponentially fast to zero, in the sense that

E(t) ≤ C 1 exp (-C 2 t) E(0), (18)
for some C 1 , C 2 > 0. The decay is uniform with respect to σ and the initial condition, that is, the constants C 1 and C 2 in ( 18) are independent of (y

(i) 0 , y (i) 1
) N i=1 and of the choice of σ verifying [START_REF] Martinez | Stabilization of the wave equation by on-off and positivenegative feedbacks[END_REF].

Proof. The proof of the first assertion can be easily obtained by adapting the arguments of Section 4.1. The proof of the second assertion is given in Section 5.

Remark 2. In the case N = 2 the star reduces to a single string. (The optimal switching control problem for the interval case N = 2 is studied in [START_REF] Gugat | Optimal switching boundary control of a string to rest in finite time[END_REF].) Neither condition [START_REF] Leugering | On exact controllability of generic trees, in "Contrôle des systèmes gouvernés par des équations aux dérivées partielles (Nancy[END_REF] nor condition [START_REF] Martinez | Stabilization of the wave equation by on-off and positivenegative feedbacks[END_REF] guarantee the exponential stabilization in this case. Indeed, it is possible to construct nonzero periodic solutions of (S) satisfying both ( 15) and ( 17): Take for simplicity c 1 = c 2 = c and L 1 = L 2 = L, in such a way that both [START_REF] Leugering | On exact controllability of generic trees, in "Contrôle des systèmes gouvernés par des équations aux dérivées partielles (Nancy[END_REF] and ( 17) provide the same condition σ 1 (t) + σ 2 (t) ≥ 1. Notice, as in [START_REF] Martinez | Stabilization of the wave equation by on-off and positivenegative feedbacks[END_REF], that there exist σ 1 and σ 2 satisfying σ 1 + σ 2 ≡ 1, piecewise constant, 2L/cperiodic, and an optic ray never touching the boundary points when the damping is active (see Figure 1). This allows to construct a periodic nonzero solution of (S) for such choice of σ 1 and σ 2 (see [START_REF] Martinez | Stabilization of the wave equation by on-off and positivenegative feedbacks[END_REF] for details).

σ 2 = 1 0 2L t x σ 1 = 1 σ 2 = 1 σ 1 = 1 σ 1 = 1 σ 2 = 1

4.1.

Velocity Decay to Zero in Finite Time. In this section we prove the first part of Theorem 3.1. So assume that for all i ∈ {1, ..., N } we have κ i = 1.

If σ i (t -Li ci ) = 1, equation [START_REF] Gugat | Boundary Feedback Stabilization by Time Delay for One-Dimensional Wave Equations[END_REF] implies that α ′ i (c t) = 0. Therefore [START_REF] Leugering | On feedback controls for dynamic networks of strings and beams and their numerical simulation[END_REF] implies that α ′ i (c i t) = 0 for all i ∈ {1, ..., N } and for almost every t in (T, T + 2λ). Due to [START_REF] Haraux | Asymptotic stability for intermittently controlled second-order evolution equations[END_REF] this yields β ′ i (-c i t) = 0 for all i ∈ {1, ..., N } and for almost every t in (T, T + 2λ). Thanks to [START_REF] Krabs | On moment theory and controllability of one-dimensional vibrating systems and heating processes[END_REF], for almost every x in (0, L i ) we have

v (i) x (T + λ, x) = 0, v (i)
t (T + λ, x) = 0. So for all t > T + 2λ, the solution v is constant.

Note in particular that if the feedback is active at all boundary nodes, that is

N i=1 σ i (t) = N
for a time interval of length greater than or equal to 3λλ min , then the state becomes constant in finite time.

Exponential Decay.

In this section we prove the second part of Theorem 3.1.

Lemma 4.1. Assume that condition [START_REF] Leugering | On exact controllability of generic trees, in "Contrôle des systèmes gouvernés par des équations aux dérivées partielles (Nancy[END_REF] is satisfied. Define f = max{ 2 N , N -2 N }. Assume that for all i ∈ {1, ..., N } we have

κ i -1 κ i + 1 < 1 -f √ N .
Define the number

F = √ N max i∈{1,...,N } κ i -1 κ i + 1 + f < 1.
Then the following inequality holds for all natural numbers k:

ess sup s>2kλ max i∈{1,...,N } {|c i β ′ i (-c i s)|} ≤ F k ess sup s∈(0,2λ) max i∈{1,...,N } |c i β ′ i (-c i s)|. ( 19
)
Moreover, for all t ≥ (2k + 1)λ the following inequality holds:

ess sup{c i |v (i) x (t, x)|, |v (i) t (t, x)| : x ∈ (0, L i ), i ∈ {1, ..., N }} (20) ≤ 1 + √ N F k ess sup s∈(0,2λ) max j∈{1,...,N } |c j β ′ j (-c j s)|.
Proof. The idea of the proof is that for all s > 2kλ we can go backwards in (0, s) until a point in the interval (0, 2λ) is reached in at least k steps of length less than or equal to 2λ. In each of these steps, the essential supremum is reduced at least by a factor F .

Define γ = max i∈{1,...,N } κi-1 κi+1 . Condition [START_REF] Leugering | On exact controllability of generic trees, in "Contrôle des systèmes gouvernés par des équations aux dérivées partielles (Nancy[END_REF] implies that for almost every t > λ there exists at most one number k ∈ {1, ..., N } with σ k (t -L k /c k ) = 0. Due to (9), we have

|α ′ k (c k t)| ≤ |β ′ k (-c k (t -2L k /c k ))|.
Moreover, for the other N -1 derivatives we have the inequality

|α ′ j (c j t)| ≤ γ|β ′ j (-c j (t -2L j /c j ))|, j = k. Define the vectors a(t) =      c 1 α ′ 1 (c 1 t) c 2 α ′ 2 (c 2 t) . . . c N α ′ N (c N t)      , b(t) =      c 1 β ′ 1 (-c 1 t) c 2 β ′ 2 (-c 2 t) . . . c N β ′ N (-c N t)      . ( 21
)
Due to [START_REF] Haraux | Asymptotic stability for intermittently controlled second-order evolution equations[END_REF] we have, for almost every t > 0,

b(t) = Aa(t).
Define by Π k : R N → R N the orthogonal projection on the k-th coordinate axis. For a vector z ∈ R n let z ∞ = max j∈{1,...,N } |z j | denote the corresponding maximum norm and let z denote the Euclidean norm. On account of the definition (6) of the matrix A we have the inequality Since the matrix A is orthogonal we have, for almost every t > 0,

AΠ k a(t) ∞ ≤ f |a k (t)| ≤ f |b k (t -2L k /c k )|,
b(t) ∞ = A(a(t) -Π k a(t)) + AΠ k a(t) ∞ ≤ A(a(t) -Π k a(t)) ∞ + AΠ k a(t) ∞ ≤ A(a(t) -Π k a(t)) + AΠ k a(t) ∞ = a(t) -Π k a(t) + AΠ k a(t) ∞ ≤ √ N a(t) -Π k a(t) ∞ + AΠ k a(t) ∞ ≤ √ N γ max j =k |b j (t -2L j /c j )| + f |b k (t -2L k /c k )| ≤ √ N γ + f max j∈{1,...,N } |b j (t -2L j /c j )| = F max j∈{1,...,N } |b j (t -2L j /c j )|.
Then for all τ 2 ≥ τ 1 ≥ 2λ we have the inequality

ess sup t∈[τ1,τ2] b(t) ∞ ≤ F ess sup t∈[τ1-2λ,τ2-2λmin] b(t) ∞ , (22)
where we recall that λ min = min{L 1 /c 1 , . . . , L N /c N }. By induction, for a natural number j and τ 2 ≥ τ 1 ≥ 2jλ we get the inequality ess sup

t∈[τ1,τ2] b(t) ∞ ≤ F j ess sup t∈[τ1-2jλ,τ2-2jλmin] b(t) ∞ .
For

τ 1 = 2jλ ≤ τ 2 , this yields ess sup t∈[2jλ,τ2] b(t) ∞ ≤ F j ess sup t∈[0,τ2-2jλmin] b(t) ∞ . (23)
Using ( 22) with τ 1 = 2λ and τ 2 = T in a similar way, for all T > 2λ we obtain the inequality ess sup

t∈[0,T ] b(t) ∞ ≤ ess sup t∈[0, T -2λmin] b(t) ∞ .
By induction, for all T ≥ 2λ this implies ess sup

t∈[0,T ] b(t) ∞ ≤ ess sup t∈[0, 2λ] b(t) ∞ . (24)
With (24), inequality (23) yields ess sup

t∈[2jλ,τ2] b(t) ∞ ≤ F j ess sup t∈[0,τ2-2jλmin] b(t) ∞ ≤ F j ess sup t∈[0, 2λ] b(t) ∞ .
Since the number τ 2 can be chosen arbitrarily large, this yields [START_REF] Valein | Stabilization of the wave equation on 1-d networks[END_REF]. Due to [START_REF] Haraux | Asymptotic stability for intermittently controlled second-order evolution equations[END_REF] and the fact that the matrix A is orthogonal we have

a(s) ∞ ≤ a(s) = b(s) ≤ √ N b(s) ∞
for almost every s > 0.

If t + x/c i ≥ 2kλ and -t + x/c i ≤ -2kλ for all i ∈ {1, . . . , N } we have

max{c i |v (i) x (t, x)|, |v (i)
t (t, x)|} ≤ max i∈{1,...,N } c i [|α ′ i (x + c i t)| + |β ′ i (x -c i t)|] ≤ 1 + √ N F k ess sup s∈(0,2λ) b(s) ∞ .
This implies inequality [START_REF] Zuazua | Switching controls[END_REF].

In order to complete the proof of Theorem 3.1 let us introduce the function

ρ(t) = ess sup{c i |v (i) x (t, x)|, |v (i)
t (t, x)| : x ∈ (0, L i ), i ∈ {1, ..., N }}. Due to [START_REF] Krabs | On moment theory and controllability of one-dimensional vibrating systems and heating processes[END_REF] we have

ρ(t) ≤ max i∈{1,...,N } ess sup{c i (|α ′ i (x + c i t)| + |β ′ i (x -c i t)|) : x ∈ (0, L i )}.
We claim that ρ satisfies

ρ(t) ≤ C 0 , t ∈ (0, 3λ), (25)
ρ(t) ≤ C 1 F k , t > (2k + 1)λ, k ∈ N, (26)
with C 0 and C 1 only depending on the initial condition (y

) N i=1 and not on the choice of σ verifying [START_REF] Leugering | On exact controllability of generic trees, in "Contrôle des systèmes gouvernés par des équations aux dérivées partielles (Nancy[END_REF].

For x ∈ (0, L i ), the values of α i (x) are given by ( 7) and the values of β i (x) are given by [START_REF] Gugat | Optimal switching boundary control of a string to rest in finite time[END_REF]. Hence the constant

C * = 2 max i∈{1,...,N } ess sup{c i |α ′ i (x)|, c i |β ′ i (x)| : x ∈ (0, L i )}
only depends on the initial condition and satisfies ρ(0) ≤ C * . Equation (9) yields the inequality

c i |α ′ i (c i t)| ≤ c i |β ′ i (2L i -c i t)| ≤ C * /2, t ∈ (L i /c i , 2L i /c i ). Hence for almost every t ∈ (0, 2L i /c i ) we have c i |α ′ i (c i t)| ≤ C * /2. Since A has the matrix norm A ∞ = 3 -4
N , the node condition [START_REF] Haraux | Asymptotic stability for intermittently controlled second-order evolution equations[END_REF] implies that

c i |β ′ i (-c i t)| ≤ 3 - 4 N max{c j |α ′ j (c j t)| : j = 1, ..., N } (27)
for almost every t > 0.

Moreover, due again to (9), we have

c i |α ′ i (c i t)| ≤ 3 - 4 N max{c j |α ′ j (c j (t -2L i /c i ))| : j = 1, ..., N }, t > 2L i /c i .
Hence, by recurrence, for all integer k and almost all t < 2kλ min ,

c i |α ′ i (c i t)| ≤ C * 2 3 - 4 N k-1
.

As a consequence, if t < 2kλ min , then, for almost every x ∈ (0, L i ),

c i |α ′ i (c i t + x)| ≤ C * 2 3 - 4 N k , c i |β ′ i (x -c i t)| ≤ C * 2 3 - 4 N k ,
where the second inequality uses (27). It follows that

ρ(t) ≤ C * 3 - 4 N k , t < 2kλ min .
Now we choose k such that 3λ < 2 kλ min and set

C 0 = C * 3 - 4 N k .
If t > (2k + 1)λ for some integer k inequality [START_REF] Zuazua | Switching controls[END_REF] implies

ρ(t) ≤ 1 + √ N F k ess sup s∈(0,2λ) max j∈{1,...,N } |c j β ′ j (-c j s)| ≤ 1 + √ N C 0 F k .
Hence we choose

C 1 = 1 + √ N C 0 .
Since ρ(t) satisfies ( 25) and (26), it can be bounded from above by

F -2 max{C 0 , C 1 } exp ln(F ) 2λ t .
5. Proof of Theorem 3.2. Let ε be a positive real number to be fixed later and define, for i = 1, ..., N ,

Ξ i =    x ∈ R N : j =i x 2 j < εx 2 i    .
Hence, Ξ i is a cone with axial symmetry with respect to the axis spanned by the i-th vector of the canonical basis of R N . Let Ξ = ∪ N i=1 Ξ i . Lemma 5.1. There exists k 1 > 0 depending only on ε such that, if x belongs to R N \ Ξ, then min i∈{1,...,N } j =i

x 2 j ≥ k 1 x 2 . ( 28
)
Proof. Let i be an index achieving the minimization in (28). Since x ∈ Ξ i , then

j =i x 2 j ≥ 1 2 j =i x 2 j + εx 2 i 2 ≥ min 1 2 , ε 2 x 2
and the lemma is proved. Let a and b be defined as in [START_REF] Zuazua | Optimal and approximate control of finite-difference approximation schemes for the 1D wave equation[END_REF]. We can consider a and b as measurable vector-valued functions defined on the entire half-line [0, +∞). Define the set Θ = {t ≥ 0 : b(t) ∈ Ξ}.

Lemma 5.2. Let k 2 ∈ (0, min{2, N -2}/N ). Then, for every ε > 0 small enough, almost every t ∈ Θ and every i = 1, ..., N ,

|a i (t)| ≥ k 2 a(t) .
(29)

Proof. Let Υ = {x ∈ R N : |x i | ≥ k 2 x for every i = 1, ..., N }.
In order to prove the lemma, we have to show that for every ε > 0 small enough and almost every t ∈ Θ, a(t) ∈ Υ.

Since A is idempotent and because of (10), a(t) ∈ AΞ for almost every t ∈ Θ. Notice that AΞ is the union of the N cones with axial symmetry with respect to the columns of A and with the same aperture as the Ξ i 's.

We have to show that for every ε > 0 small enough AΞ is contained in Υ. It suffices to notice that the boundary of Υ is invariant by multiplication by a scalar and that each vector corresponding to a column of A is in the interior of Υ. (Indeed, if x is a column of A, then x = 1 and |x i | = (N -2)/N or |x i | = 2/N .) Then for ε small enough every vector of AΞ \ {0} belongs to the interior of Υ.

In the following ε will be fixed fulfilling the smallness requirement of Lemma 5.2.

Lemma 5.3. Let T = {τ : τ ≥ λ, σ i (τ -L i /c i ) = 0 for every i = 1, . . . , N }.

If σ satisfies [START_REF] Martinez | Stabilization of the wave equation by on-off and positivenegative feedbacks[END_REF], then, for almost every t ∈ T ∩ (4λ, +∞), t -2L i /c i ∈ T for at most one i ∈ {1, ..., N }.

Proof. Assume by contradiction that for all t in a subset of positive measure of T ∩ (4λ, +∞) we have t -2L i /c i , t -2L j /c j ∈ T with i = j. In particular, for all t is such set,

σ i t -2 L j c j - L i c i = σ j t -2 L i c i - L j c j = 0.
This implies that condition [START_REF] Martinez | Stabilization of the wave equation by on-off and positivenegative feedbacks[END_REF] is not satisfied when we take as t the time t -2(L i /c i + L j /c j ) (see Figure 2). Thus [START_REF] Martinez | Stabilization of the wave equation by on-off and positivenegative feedbacks[END_REF] is not satisfied on a set of positive measure and the contradiction is reached.

Let us complete the proof of Theorem 3.2.

The time-derivative of the energy E(t) is given by [START_REF] Lagnese | Modeling, analysis and control of dynamic elastic multi-link structures[END_REF]. Notice that

Ė(t) = -4 N i=1 κ i σ i (t) (κ i σ i (t) + 1) 2 c 2 i β ′ i (L i -c i t) 2 ≤ -ν N i=1 σ i (t)c 2 i β ′ i (L i -c i t) 2 ,
where

ν = 4 min κ 1 (κ 1 + 1) 2 , . . . , κ N (κ N + 1) 2 > 0. t ∈ T 0 L i L j σ j (t -2L i /c i -L j /c j ) = 0 σ i (t -2L j /c j -L i /c i ) = 0 t -2L j /c j ∈ T σ j (t -L j /c j ) = 0 t -2L i /c i ∈ T t -2L i /c i -2L j /c j σ i (t -L i /c/c i ) = 0 Figure 2. Contradiction argument for t, t -2L i /c i , t -2L j /c j ∈ T . Let F (t) = N i=1 E t + L i c i + N i=1 E t - L i c i . Then Ḟ (t) ≤ -ν N i=1 σ i t + L i c i b i (t) 2 -ν N i=1 σ i t - L i c i c 2 i β ′ i (2L i -c i t) 2 .
Lemma 5.1 and condition [START_REF] Martinez | Stabilization of the wave equation by on-off and positivenegative feedbacks[END_REF] guarantee that if t ∈ Θ, then

Ḟ (t) ≤ -νk 1 b(t) 2 = -νk 1 a(t) 2 .
On the other hand, for almost every t ∈ Θ, (29) holds. Hence, if t ∈ Θ ∩ (λ, +∞) and σ i (t -L i /c i ) = 1 for some i ∈ {1, ..., N }, then, thanks to (9), either a(t) = 0 or κ i = 1 and

|β ′ i (2L i -c i t)| = κ i + 1 |κ i -1| |α ′ i (c i t)|, leading to Ḟ (t) ≤ -ν κ i + 1 |κ i -1| k 2 a(t) 2 .
We proved that for almost every t ∈ (λ, +∞) \ T ,

Ḟ (t) ≤ -k 3 a(t) 2 ,
with k 3 > 0 not depending on the initial condition nor on the choice of (σ 1 , . . . , σ N ) satisfying [START_REF] Martinez | Stabilization of the wave equation by on-off and positivenegative feedbacks[END_REF].

According to Lemma 5.3, moreover, for almost every t ∈ T ∩ Θ ∩ (4λ, ∞), for all but possibly one i ∈ {1, ..., N } we have

Ḟ (t -2L i /c i ) ≤ -k 3 a(t -2L i /c i ) 2 = -k 3 b(2L i /c i -t) 2 ≤ -k 3 |b i (2L i /c i -t)| 2 = -k 3 |a i (t)| 2 ≤ -k 2 2 k 3 a(t) 2 . Let G(t) = F (t) + N i=1 F (t -2L i /c i ). Then, for almost every t > 4λ, Ġ(t) ≤ -(N -1)k 2 2 k 3 a(t) 2 , (30)
where we used the inequality (N -1)k 2 2 ≤ 1.

Notice that

E(t) = N i=1 1 c i Li 0 |a i (t + x/c i )| 2 + |b i (t -x/c i )| 2 dx = N i=1 1 c i Li 0 |a i (t + x/c i )| 2 + |a i (t -x/c i )| 2 dx = N i=1 t+Li/ci t-
G(t + λ) -G(t -λ) ≤ - N -1 2N (N + 1) k 2 2 k 3 G(t + 3λ).
Hence G(t) decays exponentially to zero as t goes to infinity. Moreover, G(5λ) ≤ 2N (N + 1)E(0). Since c, k 2 , k 3 , N and λ do not depend on the initial conditions nor on σ, we have that G(t) ≤ C 1 exp (-C 2 t) E(0) with C 1 and C 2 independent of (y (i) 0 , y (i) 1 ) N i=1 and of the choice of σ verifying [START_REF] Martinez | Stabilization of the wave equation by on-off and positivenegative feedbacks[END_REF]. Inequality [START_REF] Russell | Nonharmonic Fourier series in the control theory of distributed parameter systems[END_REF] follows and this concludes the proof of Theorem 3.2.

Conclusion.

For a single string it is well known that a velocity feedback at one end with a special feedback parameter steers the solution to a constant state in finite time; the semigroup describing the corresponding solution is nilpotent. For a larger set of feedback parameters the energy decays exponentially.

In this paper we prove that a similar situation occurs for star-shaped networks with boundary feedback at all boundary nodes: For the special feedback parameter the partial derivatives of the solution vanish after finite time and the system state becomes constant.

If the feedback parameter is chosen in a neighborhood of the special parameter and the feedback is switched off at the boundary nodes in such a way that the l 1 -norm of the switching vector (σ i (• -L i /c i )) N i=1 is larger than or equal to N -1 almost always, then the partial derivatives of the solution decay exponentially fast in L ∞ . Note that the node where the control is switched off need not be constant but can vary with time.

An even stronger result is obtained considering the switching vector (σ i (• + L i /c i )) N i=1 , since in this case the exponential decay of the energy holds for all strictly positive feedback parameters.

These results may be interpreted as the following robustness property of the system: The exponential stability for the boundary feedback stabilization of a starshaped network is not destroyed if at each moment (up to suitable time-shifts) one of the feedback controllers is not active.

It is a natural question (still open, up to our knowledge) whether an analogous result holds true without considering time-shifts, that is, under the hypothesis that the l 1 -norm of the switching vector (σ i (•)) N i=1 is larger than or equal to N -1 almost always.

Figure 1 .

 1 Figure 1. Optic ray never hitting the boundary points when the damping is active.

Remark 3 . 4 .

 34 Conditions (15) and[START_REF] Martinez | Stabilization of the wave equation by on-off and positivenegative feedbacks[END_REF] in Theorems 3.1 and 3.2 cannot, in general, be relaxed by taking N -2 instead of N -1. Indeed, consider the case in which c 1 = c 2 and L 1 /L 2 ∈ Q. It is known that there exist nonzero periodic solutions to the uncontrolled wave equation on the network (with Neumann boundary conditions) which are supported on the union of the first two strings. In order to construct any such solution it suffices to consider a string of length L 1 + L 2 , parameterized on the interval [0, L 1 + L 2 ], and a periodic solution having L 1 as nodal point. Then, identifying the intervals [0, L 1 ] and [L 1 , L 1 + L 2 ] with the first and the second string of the network respectively, the extension by zero of the solution on the whole network satisfies (S) with σ 1 , σ 2 ≡ 0 (see[START_REF] Dáger | Wave propagation, observation and control in 1-d flexible multistructures[END_REF] Section 4.7] for the explicit expression of the solution). This proves the existence of solutions of (S) with N i=1 σ i (t±L i /c i ) ≡ N -2 that do not converge to any constant function. Proof of Theorem 3.1.

  where a k and b k denote the k-th component of a and b respectively.

  Li/ci |a i (s)| 2 dx ≤This and (30) imply that for almost every t > 5λ the inequalityG(t + λ) -G(tλ) ≤ -(N -1)k 2 2 k 3 E(t) holds true.By monotonicity of E and definition of G,

	t+λ
	a(s) 2 ds.
	t-λ

G(t) ≤ 2N (N + 1)E(t -3λ) so that

Acknowledgement This paper took benefit from discussions during the meeting Partial differential equations, optimal design and numerics 2009 at the Benasque Center for Science Pedro Pascual.

This work is supported by Procope DAAD D/0811409 and Égide, 20161VM. The second author was partially funded by the ANR project ArHyCo, Programme ARPEGE, contract number ANR-2008 SEGI 004 01-30011459 .