Skip to main content
SpringerLink
Log in

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

  • Published:
Acta Applicandae Mathematicae Aims and scope Submit manuscript

Abstract

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3a
Fig. 3b
Fig. 3c
Fig. 4a
Fig. 4b
Fig. 4c
Fig. 5a
Fig. 5b
Fig. 5c

Similar content being viewed by others

References

  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)

    MATH  Google Scholar 

  2. Huang, F.L.: Characteristic condition for exponential stability of linear dynamical systems in Hilbert spaces. Ann. Differ. Equ. 1, 43–56 (1985)

    MathSciNet  MATH  Google Scholar 

  3. Dager, R., Zuazua, E.: Wave Propagation, Observation and Control in 1-D Flexible Multi-Structures. Springer, Berlin (2006)

    Book  Google Scholar 

  4. Komornik, V.: Exact Controllability and Stabilization: The Multiplier Method. Wiley/Masson, Chichester/Paris (1994)

    MATH  Google Scholar 

  5. Valein, J., Zuazua, E.: Stabilization of the wave equation on 1-d networks. SIAM J. Control Optim. 48(4), 2771–2797 (2009)

    Article  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  9. Dáger, R., Zuazua, E.: Controllability of star-shaped networks of strings. C. R. Acad. Sci., Sér. 1 Math. 332(7), 621–626 (2001)

    MathSciNet  MATH  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  11. Dáger, R., Zuazua, E.: Spectral boundary controllability of networks of strings. C. R. Math. Acad. Sci. Paris 334(7), 545–550 (2002)

    Article  MathSciNet  Google Scholar 

  12. Dager, R.: Observation and control of vibrations in tree-shaped networks of strings. SIAM J. Control Optim. 43(2), 590–623 (2004)

    Article  MathSciNet  Google Scholar 

  13. Ammari, K., Jellouli, M.: Stabilization of star-shaped networks of strings. Differ. Integral Equ. 17(11), 1395–1410 (2004)

    MathSciNet  MATH  Google Scholar 

  14. Ammari, K., Jellouli, M., Khenissi, M.: Stabilization of generic trees of strings. J. Dyn. Control Syst. 11(2), 177–193 (2005)

    Article  MathSciNet  Google Scholar 

  15. Ammari, K.: Asymptotic behavior of some elastic planar networks of Bernoulli-Euler beams. Appl. Anal. 86(12), 1529–1548 (2007)

    Article  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  18. Han, Z.J., Wang, L.: Riesz basis property and stability of planar networks of controlled strings. Acta Appl. Math. 110, 511–533 (2010)

    Article  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  21. Jellouli, M.: Spectral analysis for a degenerate tree and applications. Int. J. Control 88(8), 1647–1662 (2015)

    Article  MathSciNet  Google Scholar 

  22. Han, Z.J., Zuazua, E.: Decay rates for 1-d heat-wave planar networks. Netw. Heterog. Media 11(4), 655–692 (2016)

    Article  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  24. Zhang, Y.X., Xu, G.Q.: Exponential and super stability of a wave network. Acta Appl. Math. 124(1), 19–41 (2013)

    Article  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  27. Mercier, D., Régnier, V.: Spectrum of a network of Euler-Bernoulli beams. J. Math. Anal. Appl. 337, 174–196 (2008)

    Article  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  31. von Below, J.: Classical solvability of linear parabolic equations on networks. J. Differ. Equ. 72, 316–337 (1988)

    Article  MathSciNet  Google Scholar 

  32. Naimark, M.A.: Linear Differential Operators. Ungar, New York (1967)

    MATH  Google Scholar 

  33. Pazy, A.: Semigroups of Linear Operators and Applications to Partial Differential Equations. Springer, Berlin (1983)

    Book  Google Scholar 

  34. Dunford, N., Schwartz, J.T.: Linear Operators, Part III, Spectral Operators. Interscience, New York (1971)

    MATH  Google Scholar 

  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)

    MATH  Google Scholar 

  36. Young, R.M.: An Introduction to Nonharmonic Fourier Series. Academic Press, New York (1980)

    MATH  Google Scholar 

  37. Lyubich, Y.L., Phóng, V.Q.: Asymptotic stability of linear differential equations in Banach spaces. Stud. Math. 88, 34–37 (1988)

    Article  MathSciNet  Google Scholar 

  38. Trefethen, L.N.: Spectral Methods in Matlab. SIAM, Philadelphia (2000)

    Book  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. College of Science, Civil Aviation University of China, Tianjin, 300000, China

    Ya-Xuan Zhang

  2. School of Mathematics, Tianjin University, Tianjin, 300354, China

    Zhong-Jie Han & Gen-Qi Xu

Authors
  1. Ya-Xuan Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Zhong-Jie Han
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Gen-Qi Xu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Zhong-Jie Han.

Additional information

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

This research is supported by the Natural Science Foundation of China grant NSFC-61503385, 61573252 and 11705279.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Zhang, YX., Han, ZJ. & Xu, GQ. Stability and Spectral Properties of General Tree-Shaped Wave Networks with Variable Coefficients. Acta Appl Math 164, 219–249 (2019). https://doi.org/10.1007/s10440-018-00236-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10440-018-00236-y

Keywords

Mathematics Subject Classification (2000)

Search

Navigation