Stability and Spectral Properties of General Tree-Shaped Wave Networks with Variable Coefficients

  • Ya-Xuan Zhang
  • Zhong-Jie HanEmail author
  • Gen-Qi Xu


The stability of general tree-shaped wave networks with variable coefficients under boundary feedback controls is considered. Making full use of the tree-shaped structures, we present a detailed asymptotic spectral analysis of the networks. By proposing the from-root-to-leaf calculating technique, we deduce an explicit recursive expression for the asymptotic characteristic equation and the spectral properties are further obtained. We show that the spectrum-determined-growth (SDG) condition holds. Thus the stability analysis of the closed-loop system can be completely converted to the infimum estimation of the asymptotic characteristic equation. Especially, we further show that the infimum is positive so as to obtain the exponential stability by estimating the recursive expression in from-leaf-to-root order. Some numerical simulations are presented to illustrate and support the theoretical results.


Wave network Variable coefficient Exponential stability Feedback control Recursive characteristic equation Spectrum-determined-growth (SDG) condition 

Mathematics Subject Classification (2000)

35L05 35L90 37L15 93C20 93D15 



  1. 1.
    Lagnese, J., Leugering, G., Schmidt, E.J.P.G.: Modeling, Analysis and Control of Dynamic Elastic Multi-link Structures, Systems and Control: Foundations and Applications. Birkhäuser, Boston (1994) zbMATHGoogle Scholar
  2. 2.
    Huang, F.L.: Characteristic condition for exponential stability of linear dynamical systems in Hilbert spaces. Ann. Differ. Equ. 1, 43–56 (1985) MathSciNetzbMATHGoogle Scholar
  3. 3.
    Dager, R., Zuazua, E.: Wave Propagation, Observation and Control in 1-D Flexible Multi-Structures. Springer, Berlin (2006) CrossRefzbMATHGoogle Scholar
  4. 4.
    Komornik, V.: Exact Controllability and Stabilization: The Multiplier Method. Wiley/Masson, Chichester/Paris (1994) zbMATHGoogle Scholar
  5. 5.
    Valein, J., Zuazua, E.: Stabilization of the wave equation on 1-d networks. SIAM J. Control Optim. 48(4), 2771–2797 (2009) MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Yao, P.F.: Energy decay for the Cauchy problem of the linear wave equation of variable coefficients with dissipation. Chin. Ann. Math., Ser. B 31(1), 59–70 (2010) MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Guo, B.Z., Xie, Y.: A sufficient condition on Riesz basis with parentheses of non-self-adjoint operator and application to a serially connected string system under joint feedbacks. SIAM J. Control Optim. 43(4), 1234–1252 (2004) MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Xu, G.Q., Han, Z.J., Yung, S.P.: Riesz basis property of serially connected Timoshenko beams. Int. J. Control 80(3), 470–485 (2007) MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Dáger, R., Zuazua, E.: Controllability of star-shaped networks of strings. C. R. Acad. Sci., Sér. 1 Math. 332(7), 621–626 (2001) MathSciNetzbMATHGoogle Scholar
  10. 10.
    Dáger, R., Zuazua, E.: Controllability of tree-shaped networks of vibrating strings. C. R. Acad. Sci., Sér. 1 Math. 332(12), 1087–1092 (2001) MathSciNetzbMATHGoogle Scholar
  11. 11.
    Dáger, R., Zuazua, E.: Spectral boundary controllability of networks of strings. C. R. Math. Acad. Sci. Paris 334(7), 545–550 (2002) MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Dager, R.: Observation and control of vibrations in tree-shaped networks of strings. SIAM J. Control Optim. 43(2), 590–623 (2004) MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Ammari, K., Jellouli, M.: Stabilization of star-shaped networks of strings. Differ. Integral Equ. 17(11), 1395–1410 (2004) MathSciNetzbMATHGoogle Scholar
  14. 14.
    Ammari, K., Jellouli, M., Khenissi, M.: Stabilization of generic trees of strings. J. Dyn. Control Syst. 11(2), 177–193 (2005) MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Ammari, K.: Asymptotic behavior of some elastic planar networks of Bernoulli-Euler beams. Appl. Anal. 86(12), 1529–1548 (2007) MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Han, Z.J., Xu, G.Q.: Spectrum and dynamical behavior of a kind of planar network of non-uniform strings with non-collocated feedbacks. Netw. Heterog. Media 5(2), 315–334 (2010) MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Han, Z.J., Xu, G.Q.: Output feedback stabilisation of a tree-shaped network of vibrating strings with non-collocated observation. Int. J. Control 84(3), 458–475 (2011) MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Han, Z.J., Wang, L.: Riesz basis property and stability of planar networks of controlled strings. Acta Appl. Math. 110, 511–533 (2010) MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Guo, Y.N., Chen, Y.L., Xu, G.Q., Zhang, Y.X.: Exponential stabilization of variable coefficient wave equations in a generic tree with small time-delays in the nodal feedbacks. J. Math. Anal. Appl. 395, 727–746 (2012) MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Xu, G.Q., Liu, D.Y., Liu, Y.Q.: Abstract second order hyperbolic system and applications to controlled network of strings. SIAM J. Control Optim. 47(4), 1762–1784 (2008) MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Jellouli, M.: Spectral analysis for a degenerate tree and applications. Int. J. Control 88(8), 1647–1662 (2015) MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Han, Z.J., Zuazua, E.: Decay rates for 1-d heat-wave planar networks. Netw. Heterog. Media 11(4), 655–692 (2016) MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Alabau-Boussouira, F., Perrollaz, V., Rosier, L.: Finite-time stabilization of a network of strings. Math. Control Relat. Fields 5(4), 721–742 (2015) MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Zhang, Y.X., Xu, G.Q.: Exponential and super stability of a wave network. Acta Appl. Math. 124(1), 19–41 (2013) MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Nicaise, S., Valein, J.: Stabilization of the wave equation on 1-D networks with a delay term in the nodal feedbacks. Netw. Heterog. Media 2, 425–479 (2007) MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Han, Z.J., Xu, G.Q.: Dynamical behavior of networks of non-uniform Timoshenko beams system with boundary time-delay inputs. Netw. Heterog. Media 6, 297–327 (2011) MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Mercier, D., Régnier, V.: Spectrum of a network of Euler-Bernoulli beams. J. Math. Anal. Appl. 337, 174–196 (2008) MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Ammari, K., Mercier, D., Régnier, V.: Spectral analysis of the Schrodinger operator on binary tree-shaped networks and applications. J. Differ. Equ. 259(12), 6923–6959 (2015) MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Chitour, Y., Mazanti, G., Sigalotti, M.: Stability of non-autonomous difference equations with applications to transport and wave propagation on networks. Netw. Heterog. Media 11(4), 563–601 (2016) MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Wang, J.M., Guo, B.Z.: Riesz basis and stabilization for the flexible structure of a symmetric tree-shaped beam network. Math. Methods Appl. Sci. 31, 289–314 (2008) MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    von Below, J.: Classical solvability of linear parabolic equations on networks. J. Differ. Equ. 72, 316–337 (1988) MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Naimark, M.A.: Linear Differential Operators. Ungar, New York (1967) zbMATHGoogle Scholar
  33. 33.
    Pazy, A.: Semigroups of Linear Operators and Applications to Partial Differential Equations. Springer, Berlin (1983) CrossRefzbMATHGoogle Scholar
  34. 34.
    Dunford, N., Schwartz, J.T.: Linear Operators, Part III, Spectral Operators. Interscience, New York (1971) zbMATHGoogle Scholar
  35. 35.
    Avdonin, S.A., Ivanov, S.A.: Families of Exponentials. The Method of Moments in Controllability Problems for Distributed Parameter Systems. Cambridge University Press, Cambridge (1995) zbMATHGoogle Scholar
  36. 36.
    Young, R.M.: An Introduction to Nonharmonic Fourier Series. Academic Press, New York (1980) zbMATHGoogle Scholar
  37. 37.
    Lyubich, Y.L., Phóng, V.Q.: Asymptotic stability of linear differential equations in Banach spaces. Stud. Math. 88, 34–37 (1988) MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Trefethen, L.N.: Spectral Methods in Matlab. SIAM, Philadelphia (2000) CrossRefzbMATHGoogle Scholar

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© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.College of ScienceCivil Aviation University of ChinaTianjinChina
  2. 2.School of MathematicsTianjin UniversityTianjinChina

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