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The exponential decay rate of generic tree of 1-d wave equations with boundary feedback controls

  • In this paper, we study the exponential decay rate of generic tree of 1-d wave equations with boundary feedback controls. For the networks, there are some results on the exponential stability, but no result on estimate of the decay rate. The present work mainly estimates the decay rate for these systems, including signal wave equation, serially connected wave equations, and generic tree of 1-d wave equations. By defining the weighted energy functional of the system, and choosing suitable weighted functions, we obtain the estimation value of decay rate of the systems.

    Citation: Yaru Xie, Genqi Xu. The exponential decay rate of generic tree of 1-d wave equations with boundary feedback controls[J]. Networks and Heterogeneous Media, 2016, 11(3): 527-543. doi: 10.3934/nhm.2016008

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  • In this paper, we study the exponential decay rate of generic tree of 1-d wave equations with boundary feedback controls. For the networks, there are some results on the exponential stability, but no result on estimate of the decay rate. The present work mainly estimates the decay rate for these systems, including signal wave equation, serially connected wave equations, and generic tree of 1-d wave equations. By defining the weighted energy functional of the system, and choosing suitable weighted functions, we obtain the estimation value of decay rate of the systems.


    There are still many issues related to children's health, such as lack of time for physical activity, increased time with screens, sedentary time and unbalanced diets [1][3]. It has been reported that a healthy lifestyle in childhood is associated with disease risk and mental risk in adulthood [4], and many studies have been conducted on how to promote and acquire healthy lifestyles from childhood [5]. Children's healthy lifestyles and health behaviors have been known to be affected by various factors [6],[7]. Family environment and parents/guardians are influential on children's lifestyles [5],[8], of which parental health literacy is likely to be a strongly relevant key factor [6],[9],[10]. Because children depend on their parents and guardians to prevent and deal with health problems, this suggests that children may be disadvantaged if their parents' and guardians' knowledge and skills, that is, their health literacy, are inadequate [6]. Although there are previous reports on parents'/guardians' health literacy and children's physical activity, nutrient status and screen time, it seems that reports on other patterns of behaviors and the amount of time children spend at home are limited. Therefore, this study examined the association between children's typical lifestyle behaviors, spending time at home and health literacy of their parent/guardian by a cross-sectional study.

    We conducted a cross-sectional questionnaire survey on the lifestyle of schoolchildren and their guardians between November 2015 and March 2016. Nine public elementary schools covering all grades of children (grades 1st to 6th, aged 6–12) in regional central cities in Northern and Southern districts in Japan participated, with a total of 3327 guardians among 4263 enrollees (cooperation rate: 78.0%).

    The question about time spent at home by the child asked for average times (in minutes) per day on a usual day and noted separately weekends and weekdays. The following 7 items were included: watching television, including DVDs and/or another video; playing video games (including handy-type); studying, including homework; reading books (including comic book reading, except for homework); help with family and housework; outside playing; and time doing hobbies.

    To estimate health literacy (HL), we used a validated questionnaire with five items, which was short and adapted to the Japanese population [11]. The questions asked about the degree to which a person (i) can gather information about one's own illness and health from various sources, such as newspapers, books, television and the internet; (ii) can pick out the information one needs from lots of information; (iii) can understand the information and communicate it to others; (iv) can judge the credibility of the information; and (v) can decide on plans and actions to improve one's own health based on the information. The structures of these questions are based on communicative HL for the first three questions (items i–iii) and critical HL for the latter two (items iv–v). Each item was rated on a 5-point Likert scale ranging from 1 (“strongly disagree / not at all”) to 5 (“strongly agree”), where high points mean high literacy. The points were summed, and the total scores ranged from a minimum of 5 to a maximum of 25 points.

    The questionnaire also contained basic characteristics of the parent/guardian (position from child's side, age, type of employment) and family environment (number of family, number of children, source of household income, feeling of financial leeway). As school policy limited detailed questions such as child's age, school grade, sex/gender and parents'/guardians' personal status, such as marital status, educational background, home economics and other social indicators, alternative minimum questions were adopted in this study.

    We excluded imperfect responses for child's time spent and items of health literacy, and 3188 individual data were used in analyses (4.2% of the participants excluded). Variables are presented as mean ± standard deviation for continuous variables or prevalence (%) for categorical variables. The total HL score was classified into two categories based on the median or average score. This classification has been adopted in similar previous studies for Japanese people [12][14]. In this study, the cut-off score was 18, with less than 18 as the low group (low HL) and 18 or more as the high group (high HL).

    We used the chi-square test for comparisons of proportions and Welch's t-test for continuous variables between two groups (low HL, high HL). The distribution of the data showed that a 30-minute interval was appropriate. We also calculated the odds ratio (OR) and the 95% confidence intervals (95% CI) using logistic regression analysis for each category of child time spent in the high HL group at 30 minutes or more with less than 30 minutes as a reference. Adjusting variables included the following: position of parent/guardian from the child's side, age group of parent/guardian, type of employment of parent/guardian, number of family members, number of children, major source of household income, feeling of financial leeway.

    All statistical analyses were performed using SPSS version 25 for Windows (IBM Corp., Chicago, IL, USA). The level of statistical significance for each analysis was set at P < 0.05.

    This survey was conducted according to the Ethical Guidelines for Epidemiological Studies established by the Ministry of Health, Labor and Welfare & Ministry of Education, Culture, Sports, Science and Technology in Japan. The Ethics Committee Tohoku University Graduate School of Medicine approved the research protocol (No. 2015–1–810, 2019–1–482). The survey was anonymous, and the submission of the questionnaire was regarded as consent to participate.

    Table 1 shows the background characteristics of parents/guardians and family environments. The overall characteristics of the parents/guardians who participated in this study were the following: Most were mothers (95.2%), and most were in their 30s–40s (47.4%). The most common type of employment was part-time (46.6%), and the next most common was housewife (23.1%).

    The most common number of family members was 3–5 (83.1%), with two children (49.9%) being the most common. The main source of household income was a full-time work-based company salary (71.1%), and almost half (49.5%) of the respondents answered that they felt they did not have enough financial leeway. Comparing the characteristics of the HL groups, significant differences were found in the dispersion trends for position from the child's side, number of family members, major source of household income and feeling of financial leeway.

    Table 2 shows the results of comparison of child time spent in minutes (min) for the seven behavioral categories by parent/guardian HL group. The child's time spent on watching television was longer in the group with low HL on both weekdays and weekends, with statistically significant differences between the two groups (weekday: low HL group 90.3 min, high HL group 83.2 min, difference 7.07 min, P < 0.001; similarly, weekend: 146.9 min, 136.3 min, difference 10.67 min, P = 0.001). The child's time spent on playing video games was also longer in the group with low HL on both weekdays and weekends, with statistically significant differences between the two groups (weekday: low HL group 34.7 min, high HL group 30.1 min, difference 4.62 min, P < 0.001; similarly, weekend: 70.5 min, 62.0 min, difference 8.51 min, P = 0.001). There were no significant differences between HL groups for time spent studying, reading books or helping with family/housework. The tendency was for time of studying and helping family/housework to be slightly longer in the higher HL group and time of reading books to be slightly longer in the lower HL group. The child's time spent on playing outside was greater in the high HL group both on weekdays and on weekends. A statistically significant difference was shown only on weekdays (low HL group 28.8 min, high HL group 32.3 min, difference 3.50 min, P < 0.001). The child's time spent on doing hobbies was greater in the high HL group on both weekdays and weekends, with statistically significant difference (weekday: low HL group 8.4 min, high HL group 10.0 min group, difference 1.64 min, P < 0.038; similarly, weekend: 17.8 min, 21.6 min, difference 3.74 min, P = 0.014).

    Table 1.  Demographic characteristics among parents/guardians according to HL groups.
    Total
    (n = 3188)
    Parent/guardian HL score
    P*
    Low group
    (n = 1689)
    High group
    (n = 1499)
    Position from child's side
    Mother 95.2 94.6 95.8 0.035
    Father 3.5 3.8 3.2
    Others 0.8 1.2 0.4
    Age group, years old
    <30 2.4 2.1 2.7 0.351
    30–39 47.4 47.0 47.8
    40–49 47.2 47.7 46.7
    >49 2.5 2.8 2.2
    Type of employment
    Full time job 22.0 23.1 22.5 0.114
    Part time job 46.6 43.4 45.1
    Self-employed 4.0 4.9 4.4
    Housewife 23.1 24.5 23.8
    On leave of absence/parental leave 0.9 1.4 1.2
    Seeking employment 0.4 0.4 0.4
    Others 2.4 1.3 1.9
    Number of family members
    <3 persons 2.4 2.8 1.8 0.003
    3–5 persons 83.1 81.1 85.5
    >5 persons 14.1 15.7 12.1
    Unknown 0.5 0.4 0.6
    Number of children
    1 child 15.0 14.8 15.3 0.896
    2 children 49.9 49.8 50.0
    >2 children 35.1 35.4 34.8
    Major source of household income
    Self-employed (including agriculture, forestry, and fisheries) 7.6 7.7 7.4 0.045
    Company employee, full-time salary 71.1 72.2 70.0
    Company employee, part-time salary 12.6 11.1 14.2
    Public servant salary 5.1 4.9 5.3
    Others 0.4 0.5 0.3
    Feeling of financial leeway
    Enough 26.6 22.9 30.9 <0.001
    Not enough 49.5 51.5 47.4
    Neither 22.8 24.3 21.1

    *Note: *Comparisons of variance between groups of HL; “Unknown” responses were removed from the table; total % is less than 100 in some places.

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    Table 2.  Contents and minutes of child's time spent at home compared by HL groups.
    Parent/guardian HL score
    Difference* P#
    Low group High group
    Contents of daily spending time
    Watching television
    Weekday 90.3±57.4 83.2±54.3 7.07 <0.001
    Weekend 146.9±94.5 136.3±90.2 10.67 0.001
    Playing video games
    Weekday 34.7±39.6 30.1±34.4 4.62 <0.001
    Weekend 70.5±72.1 62.0±68.0 8.51 0.001
    Studying
    Weekday 49.2±25.7 49.3±25.4 −0.11 0.902
    Weekend 42.1±39.0 43.3±35.5 −1.22 0.353
    Reading books
    Weekday 18.2±22.6 17.6±19.4 0.64 0.389
    Weekend 26.9±32.9 26.1±30.8 0.83 0.463
    Helping with family/housework
    Weekday 10.5±12.6 11.0±12.2 −0.47 0.280
    Weekend 15.6±20.8 15.8±17.6 −0.19 0.781
    Playing outside
    Weekday 28.8±35.5 32.3±36.6 −3.50 0.006
    Weekend 73.4±85.9 78.0±89.7 −4.65 0.136
    Doing hobbies
    Weekday 8.4±22.6 10.0±22.0 −1.64 0.038
    Weekend 17.8±40.7 21.6±44.7 −3.74 0.014

    *Note: Values in table: mean ± standard deviation, values in minutes; *Difference of the mean of the low-scoring group minus the mean of the high-scoring group; #Welch t-test.

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    Table 3.  Results of odds ratios of time spent by children with high HL group parents/guardians.
    No. in HL group
    Crude OR
    (95%CI)
    P Adjusted OR
    (95%CI)
    P
    Low High
    Watching television
    Weekday <30 min 87 107 1.00 (ref.) 1.00 (ref.)
    ≥30 min 1602 1392 0.71 (0.53–0.95) 0.020 0.71 (0.53–0.95) 0.022
    Weekend <30 min 104 108 1.00 (ref.) 1.00 (ref.)
    ≥30 min 1585 1391 0.85 (0.64–1.12) 0.237 0.86 (0.65–1.14) 0.304
    Playing video games
    Weekday <30 min 729 701 1.00 (ref.) 1.00 (ref.)
    ≥30 min 960 798 0.86 (0.75–0.99) 0.041 0.86 (0.75–0.99) 0.038
    Weekend <30 min 413 440 1.00 (ref.) 1.00 (ref.)
    ≥30 min 1276 1059 0.78 (0.67–0.91) 0.002 0.78 (0.66–0.91) 0.002
    Studying
    Weekday <30 min 165 133 1.00 (ref.) 1.00 (ref.)
    ≥30 min 1520 1362 1.11 (0.87–1.41) 0.387 1.11 (0.87–1.41) 0.396
    Weekend <30 min 474 378 1.00 (ref.) 1.00 (ref.)
    ≥30 min 1214 1121 1.16 (0.99–1.36) 0.068 1.16 (0.99–1.36) 0.064
    Reading books
    Weekday <30 min 1111 988 1.00 (ref.) 1.00 (ref.)
    ≥30 min 576 511 1.00 (0.86–1.16) 0.974 0.99 (0.86–1.15) 0.904
    Weekend <30 min 903 779 1.00 (ref.) 1.00 (ref.)
    ≥30 min 786 720 1.06 (0.92–1.22) 0.399 1.05 (0.91–1.21) 0.482
    Helping with family/housework
    Weekday <30 min 1465 1296 1.00 (ref.) 1.00 (ref.)
    ≥30 min 223 203 1.03 (0.84–1.26) 0.784 1.05 (0.86–1.29) 0.632
    Weekend <30 min 1280 1124 1.00 (ref.) 1.00 (ref.)
    ≥30 min 409 375 1.04 (0.89–1.23) 0.600 1.06 (0.90–1.25) 0.486
    Playing outside
    Weekday <30 min 864 683 1.00 (ref.) 1.00 (ref.)
    ≥30 min 825 816 1.25 (1.09–1.44) 0.002 1.24 (1.08–1.43) 0.003
    Weekend <30 min 524 425 1.00 (ref.) 1.00 (ref.)
    ≥30 min 1165 1074 1.14 (0.98–1.32) 0.100 1.14 (0.97–1.33) 0.105
    Doing hobbies
    Weekday <30 min 1445 1208 1.00 (ref.) 1.00 (ref.)
    ≥30 min 244 291 1.43 (1.18–1.72) <0.001 1.39 (1.15–1.68) 0.001
    Weekend <30 min 1270 1049 1.00 (ref.) 1.00 (ref.)
    ≥30 min 419 450 1.30 (1.11–1.52) 0.001 1.27(1.09–1.49) 0.003

    *Note: Abbreviations in the table: OR = odds ratio, CI = confidence interval, ref. = reference; Adjustment variables as follows: position of parent/guardian from child's side, age group of parent/guardian, type of employment of parent/guardian, number of family members, number of children, major source of household income, feeling of financial leeway.

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    Table 3 shows the odds ratios for child time spent for more than 30 minutes compared to less than 30 minutes in the group with high parental/guardian HL. The ORs of spending more than 30 minutes for watching television and video game playing were lower in the higher HL group. The adjusted ORs were 0.71 (95% CI = 0.53–0.95, P = 0.002) for weekday television watching, 0.86 (95% CI = 0.75–0.99, P = 0.038) for weekday video game playing and 0.78 (95% CI = 0.66–0.91, P = 0.002) for weekend video game playing. The ORs were close to 1.00 for study time, reading time and helping with family/housework either on weekdays or weekends. The ORs of spending more than 30 minutes playing outside and doing hobbies were higher in the higher HL group. The adjusted ORs were 1.24 (95% CI = 1.08–1.43, P = 0.003) for weekday playing outside, 1.39 (95% CI = 1.15–1.68, P = 0.001) for weekday hobby time and 1.27 (95% CI = 1.09–1.49, P = 0.003) for weekend hobby time. In a detailed analysis, significant differences in weekday television watching, weekday playing video games, weekend playing video games and weekday outside playing were even more noticeable for the more than 120 minutes time (not shown in the result table). The adjusted ORs were 0.65 (95% CI = 0.46–0.92, P = 0.041) for weekday television watching, 0.64 (95% CI = 0.47–0.88, P = 0.006) for weekday playing video games, 0.67 (95% CI = 0.55–0.82, P < 0.001) for weekend playing video games and 1.46 (95% CI = 1.05–2.04, P = 0.026) for weekday outside playing.

    We examined the association between parental/guardian health literacy and child's time spent at home, based on data of approximately 3000 Japanese schoolchildren. Overall, parental/guardian HL was strongly associated with the following four behavioral categories of time spent by children: watching television, playing video games, playing outside and doing hobbies. Our results for TV time and game time are consistent with previous studies, and regarding outdoor play as a physical activity, the direction of the results of the present study agrees with other previous studies [1],[3],[6][8].

    The largest difference in terms of number of hours was found for weekend watching television, which was approximately 10.6 minutes/day longer in the low parent/guardian HL group. The second largest time difference was found for weekend time playing video games, which was about 8.5 minutes/day longer in the low parent/guardian HL group. Children in the high parent/guardian HL group spent approximately 4.6 minutes more time playing outside on weekends and 3.7 minutes more time doing hobbies on weekends than children in the low parent/guardian HL group. The differences were measured in minutes and appeared small; however, this difference would be larger with long-term cumulation. Elementary schools in Japan have roughly 200 school days in a year. It is estimated that a difference of 7 min/day in children's weekday TV watching amounts to a difference of 1400 min/year, that is, about 23 hours.

    The results of the logistic analysis indicated that higher parental/guardian HL may reduce the probability of children spending more than 30 minutes watching TV on weekdays and playing games on weekdays and weekends. Higher parent/guardian HL indicated that the probability of children spending more than 30 minutes outside on weekdays and on weekday and weekend hobbies increased. Sub-analyses also suggested that higher parental/guardian HL, less likely for children to spend more than 2 hours watching TV on weekdays and playing games on weekdays and weekends while more likely for children to spend more than 2 hours playing outside. These results suggest that if parents/guardians have a high level of HL, it might be possible to reduce the amount of time children spend watching TV and playing games to less than 30 minutes or less than 120 minutes and to increase the amount of time they spend playing outside and enjoying their hobbies to more than 30 minutes or more than 120 minutes.

    In recent years, reports have also accumulated on children's screen time, sedentary time and their health disadvantages [3],[5],[15],[16]. The TV watching and video gaming time collected in this study could be substituted as total screen time or sitting time, although use of computers, mobile phones and smartphones was not included. Hence, screen time and sitting time were not estimated in this study. Our study also has some limitations. First, the data on children's time spent at home was based on questionnaire responses by parents/guardians and not actual measurement data, so some response errors were unavoidable. There may also be differences in content-specific time by child age and by sex. Our present study has not fully examined the variables. The information such as the child's sex/gender, year of age and class could not be added to the questionnaire due to concerns about the possibility of identifying the child. In particular, the time spent playing outside may differ by gender and age group. This remains an issue for future surveys. It has been also reported that parenting style could be significantly related to children's daily life and health behaviors [17][20]. The present study may not have adequately adjusted for parental parenting attitudes. It may also be insufficient for SES indicators, but the impact of SES as a predictor of health literacy appears to be limited [21].

    The present study focused on the relationship between parental/guardian HL and seven behavioral categories of child's time spent at home. There are few reports investigating the time spent by Japanese schoolchildren and the parents' HL, so the results of this study are very meaningful. Many factors, including the parental/guardian HL focused on in this study, have direct and indirect or combinatory influences on children's health behaviors and health outcomes. Further findings are expected from comprehensive and continuing epidemiological study.

    Our study conducted in Japanese schoolchildren showed that parental/guardian HL was associated with some contents of child's time spent at home. High parental/guardian HL was negatively associated with children's time spent watching TV and playing games, while times spent playing outside and doing hobbies were positively associated.

    [1] K. Ammari and M. Jellouli, Stabilization of star-shaped tree of elastic strings, Differential & Integral Equations, 17 (2004), 1395-1410.
    [2] K. Ammari, M. Jellouli and M. Khenissi, Stabilization of generic trees of strings, Journal of Dynamical & Control Systems, 11 (2005), 177-193. doi: 10.1007/s10883-005-4169-7
    [3] K. Ammari and D. Mercier, Boundary feedback stabilization of a chain of serially connected strings, Evol. Equ. Control Theory, 4 (2015), 1-19. doi: 10.3934/eect.2015.4.1
    [4] K. Ammari, D. Mercier and V. Régnier, Spectral analysis of the Schrodinger operator on binary tree-shaped networks and applications, Journal of Differential Equations, 259 (2015), 6923-6959. doi: 10.1016/j.jde.2015.08.017
    [5] K. Ammari, A. Henrot and M. Tucsnak, Asymptotic behaviour of the solutions and optimal location of the actuator for the pointwise stabilization of a string, Asymptotic Analysis, 28 (2001), 215-240.
    [6] K. Bartecki, A general transfer function representation for a class of hyperbolic distributed parameter systems, International Journal of Applied Mathematics & Computer Science, 23 (2013), 291-307. doi: 10.2478/amcs-2013-0022
    [7] A. V. Balakrishnan, On superstable semigroup of operators, Dynamic Systems & Applications, 5 (1996), 371-384.
    [8] J. M. Coron, B. D'Andrea-Novel and G. Bastin, A lyapunov approach to control irrigation canals modeled by the Saint Venant equations, Control Conference. IEEE, 1999.
    [9] J. M. Coron, J. De Halleux and G. Bastin, On boundary control design for quasi-linear hyperbolic systems with entropies as Lyapunov functions, Proceedings of the IEEE Conference on Decision and Control, 3 (2003), 3010-3014.
    [10] J. M. Coron, B. d'Andrea-Novel and G. Bastin, A strict Lyapunov function for boundary control of hyperbolic systems of conservation laws, Proceedings of the IEEE Conference on Decision & Control, 52 (2007), 2-11. doi: 10.1109/TAC.2006.887903
    [11] R. F. Curtain and H. J. Zwart, An Introduction to Infinite-Dimensional Linear Systems Theory, Springer-Verlag, New York, Berlin, Heidelberg, 1995. doi: 10.1007/978-1-4612-4224-6
    [12] S. Cox and E. Zuazua, The rate at which energy decays in a damped string, Communication Partial Differential Equations, 19 (1994), 213-243. doi: 10.1080/03605309408821015
    [13] R. Dager and E. Zuazua, Controllability of star-shaped networks of strings, C. R. Acad. Sci. Paris, Series I, 332 (2001), 621-626. doi: 10.1016/S0764-4442(01)01876-6
    [14] R. Dager, Observation and control of vibrations in tree-shaped networks of strings, SIAM J. Control & Optim, 43 (2004), 590-623. doi: 10.1137/S0363012903421844
    [15] R. Dager and E. Zuazua, Wave Propagation, Observation and Control in 1-d Flexible Multi-Structures, Springer Berlin, 2006. doi: 10.1007/3-540-37726-3
    [16] A. Diagne, G. Bastin and J. M. Coron, Lyapunov exponential stability of 1-D linear hyperbolic systems of balance laws, Automatica, 48 (2012), 109-114. doi: 10.1016/j.automatica.2011.09.030
    [17] Y. N. Guo and G. Q. Xu, Stability and Riesz basis property for general network of strings, Journal of dynamical & control systems, 15 (2009), 223-245. doi: 10.1007/s10883-009-9064-1
    [18] Y. N. Guo and G. Q. Xu, Exponential stabilization of a tree shaped network of strings with variable coefficients, Glasgow Mathematical Journal, 53 (2011), 481-499. doi: 10.1017/S0017089511000085
    [19] Y. N. Guo and G. Q. Xu, Exponential stabilization of variable coefficient wave equations in a generic tree with small time-delays in the nodal feedbacks, Journal of Mathematical Analysis & Applications, 395 (2012), 727-746. doi: 10.1016/j.jmaa.2012.05.079
    [20] B. Z. Guo and Y. Xie, A sufficient condition on Riesz basis with parentheses of non-selfadjoint operator and application to a serially connected string system under joint feedbacks, SIAM journal on control & optimization, 43 (2004), 1234-1252. doi: 10.1137/S0363012902420352
    [21] M. Gugat and M. Herty, Existence of classical solutions and feedback stabilization for the flow in gas networks, ESAIM Control Optimisation & Calculus of Variations, 17 (2011), 28-51. doi: 10.1051/cocv/2009035
    [22] F. L. Huang, Characteristic conditions for exponential stability of linear dynamical systems in Hilbert spaces, Ann. Differential Equations, 1 (1985), 43-56.
    [23] Z. J. Han and L. Wang, Riesz basis property and stability of planar networks of controlled strings, Acta Applicandae Mathematicae, 110 (2010), 511-533. doi: 10.1007/s10440-009-9459-8
    [24] Z. J. Han and G. Q. Xu, Spectrum and dynamical behavior of a kind of planar network of non-uniform strings with non-collocated feedbacks, Networks & Heterogeneous Media, 5 (2010), 315-334. doi: 10.3934/nhm.2010.5.315
    [25] Z. J. Han and G. Q. Xu, Stabilization and SDG condition of serially connected vibrating strings system with discontinuous displacement, Asian Journal of Control, 14 (2012), 95-108. doi: 10.1002/asjc.218
    [26] M. Jellouli, Spectral analysis for a degenerate tree and applications, International Journal of Control, 88 (2015), 1647-1662. doi: 10.1080/00207179.2015.1012652
    [27] M.Krstic, B. Z. Guo and A. Balogh, Output-feedback stabilization of an unstable wave equation, Automatica, 44 (2008), 63-74. doi: 10.1016/j.automatica.2007.05.012
    [28] J. E. Lagnese, G. Leugering and E. J. P. G. Schmidt, Modelling, Analysis and Control of Dynamic Elastic Multi-Link Structures, Systems & Control: Foundations & Applications, 1994. doi: 10.1007/978-1-4612-0273-8
    [29] J. E. Lagnese, Recent progress and open problems in control of multi-link elastic structures, Contemp. Math, 209 (1997), 161-175. doi: 10.1090/conm/209/02765
    [30] J. L. Lions, Exact controllability, stabilization and perturbations for distributed parameter system, SIAM Review, 30 (1988), 1-68. doi: 10.1137/1030001
    [31] K. S. Liu, F. L. Huang and G. Chen, Exponential stability analysis of a long chain of coupled vibrating strings with dissipative linkage, SIAM Journal on Applied Mathematics, 49 (1989), 1694-1707. doi: 10.1137/0149102
    [32] G. Leugering and E. Zuazua, On exact controllability of generic trees, Esaim Proceedings, 8 (2000), 95-105. doi: 10.1051/proc:2000007
    [33] G. Leugering, Dynamic domain decomposition of optimal control problems for networks of strings and Timoshenko beams, SIAM Journal on Control & Optimization, 37 (1999), 1649-1675. doi: 10.1137/S0363012997331986
    [34] D. Y. Liu, Y. F. Shang and G. Q. Xu, Design of controllers and compensators of a serially connected string system and its Riesz basis, Kongzhi Lilun Yu Yinyong/Control Theory & Applications, 5 (2008), 815-818.
    [35] M. Najafi, G. R. Sarhangi and H. Wang, Stabilizability of coupled wave equations in parallel under various boundary conditions, Automatic Control IEEE Transactions on, 42 (1997), 1308-1312. doi: 10.1109/9.623099
    [36] S. Nicaise and J. Valein, Stabilization of the wave equation on 1-D networks with a delay-term in the feedbacks, Networks & Heterogeneous Media, 2 (2007), 425-479. doi: 10.3934/nhm.2007.2.425
    [37] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Berlin, Germany: Springer-Verlag, 1983. doi: 10.1007/978-1-4612-5561-1
    [38] D. L. Rusell, Mathematical models for the elastic beam and their control-theoretic implications, Semigroups, Theory and Applications, 152 (1986), 177-216.
    [39] S. Rolewicz, On controllability of systems of strings, Studia Math, Studia Mathematica, 36 (1970), 105-110.
    [40] Y. F. Shang, D. Y. Liu and G. Q. Xu, Super-stability and the spectrum of one-dimensional wave equations on general feedback controlled networks, IMA Journal of Mathematical Control & Information, 31 (2014), 73-99. doi: 10.1093/imamci/dnt003
    [41] M. A. Shubov, Basis property of eigenfunctions of nonselfadjoint operator pencils generated by the equations of nonhomogeneous damped string, Integral Equations & Operator Theory, 25 (1996), 289-328. doi: 10.1007/BF01262296
    [42] E. J. P. G. Schmidt, On the modelling and exact controllability of networks of vibrating strings, Siam Journal on Control & Optimization, 30 (1992), 229-245. doi: 10.1137/0330015
    [43] L. Wang, Z. J. Han and G. Q. Xu, Exponential stability of serially connected thermoelastic system of type II with nodal damping, Applicable Analysis, 93 (2014), 1495-1514. doi: 10.1080/00036811.2013.836596
    [44] H. Wang and G. Q. Xu, Exponential stabilization of 1-d wave equation with input delay, Wseas Transactions on Mathematics, 10 (2013), 1001-1013.
    [45] G. Q. Xu, S. P. Yung and L. K. Li, Stabilization of wave systems with input delay in the boundary control, ESAIM Control Optimisation & Calculus of Variations, 12 (2006), 770-785. doi: 10.1051/cocv:2006021
    [46] G. Q. Xu, Stabilization of string system with linear boundary feedback, Nonlinear Analysis Hybrid Systems, 1 (2007), 383-397. doi: 10.1016/j.nahs.2006.07.003
    [47] G. Q. Xu and B. Z. Guo, Riesz basis property of evolution equations in Hilbert spaces and application to a coupled string equation, SIAM Journal on Control & Optimization, 42 (2003), 966-984. doi: 10.1137/S0363012901400081
    [48] G. Q. Xu, D. Y. Liu and Y. Q. Liu, Abstract second order hyperbolic system and applications to controlled network of strings, SIAM Journal on Control & Optimization, 47 (2008), 1762-1784. doi: 10.1137/060649367
    [49] G. Q. Xu and N. E. Mastorakis, Differential Equations on Metric Graph, Athens: World Scientific and Engineering Academy and Society, 2010.
    [50] G. Q. Xu and Y. X. Zhang, The exponential stability of complex differential networks, Journal of Systems Science & Mathematical Sciences, 29 (2009), 1399-1418.
    [51] E. Zuazua, Control and stabilization of waves on 1-d networks, Modelling and Optimisation of Flows on Networks, Springer Berlin Heidelberg, 2062 (2013), 463-493. doi: 10.1007/978-3-642-32160-3_9
    [52] Y. X. Zhang and G. Q. Xu, Controller design for Bush-type 1-D wave neworks, ESAIM Control Optimisation & Calculus of Variations, 18 (2012), 208-228. doi: 10.1051/cocv/2010050
    [53] Y. X. Zhang and G. Q. Xu, A new approach for the stability analysis of wave networks, Abstract and Applied Analysis, 2014 (2014), Art. ID 724512, 10 pp. doi: 10.1155/2014/724512
    [54] Y. X. Zhang and G. Q. Xu, Exponential and super stability of a wave network, Acta Applicandae Mathematicae An International Survey Journal on Applying Mathematics & Mathematical Applications, 124 (2013), 19-41. doi: 10.1007/s10440-012-9768-1
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