Exact Controllability of the 1-D Wave Equation on Finite Metric Tree Graphs

  • Sergei Avdonin
  • Yuanyuan ZhaoEmail author


In this paper, we consider initial boundary value problems and control problems for the wave equation on finite metric graphs with Dirichlet boundary controls. We propose new constructive algorithms for solving initial boundary value problems on general graphs and boundary control problems on tree graphs. We demonstrate that the wave equation on a tree is exactly controllable if and only if controls are applied at all or all but one of the boundary vertices. We find the minimal controllability time and prove that our result is optimal in the general case. The proofs for the shape and velocity controllability are purely dynamical, while the proof for the full controllability utilizes both dynamical and moment method approaches.


Boundary control Controllability Tree graphs Wave equation 

Mathematics Subject Classification

35L05 35Q93 93B05 93C20 



The authors would like to thank the referees for their valuable comments which helped to improve the manuscript.


  1. 1.
    Ali Mehmeti, F.: Nonlinear waves in networks. Math. Res. 80, 171 (1994)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Ali Mehmeti, F., Meister, E.: Regular solutions of transmission and interaction problems for wave equations. Math. Methods Appl. Sci. 11(5), 665–685 (1989)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Arioli, M., Benzi, M.: A finite element method for quantum graphs. IMA J. Numer. Anal. 38(3), 1119–1163 (2017)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Avdonin, S.: Control problems on quantum graphs. In: Analysis on Graphs and Its Applications. Proceedings of Symposia in Pure Mathematics. AMS, vol. 77, pp. 507–521 (2008)Google Scholar
  5. 5.
    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
  6. 6.
    Avdonin, S., Kurasov, P.: Inverse problems for quantum trees. Inverse Probl. Imaging 2(1), 1–21 (2008)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Avdonin, S., Mikhaylov, V.: The boundary control approach to inverse spectral theory. Inverse Probl. 26(4), 045009 (2010)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Avdonin, S., Nicaise, S.: Source identification problems for the wave equation on graphs. Inverse Probl. 31(9), 095007 (2015)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Avdonin, S., Edward, J.: Exact controllability for string with attached masses. SIAM J. Control Optim. 56(2), 945–980 (2018)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Avdonin, S., Edward, J.: Controllability for a string with attached masses and Riesz bases for asymmetric spaces. Math. Control Relat. Fields 9(3), 453–494 (2019)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Belishev, M., Vakulenko, A.: Inverse problems on graphs: recovering the tree of strings by the bc-method. J. Inverse Ill Posed Probl. 14(1), 29–46 (2006)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Bergeron, F., Labelle, G., Leroux, P.: Combinatorial Species and Tree-Like Structures, vol. 67. Cambridge University Press, Cambridge (1998)zbMATHGoogle Scholar
  13. 13.
    Dáger, R., Zuazua, E.: Wave Propagation, Observation and Control in 1-d Flexible Multi-structures, vol. 50. Springer, Berlin (2006)CrossRefGoogle Scholar
  14. 14.
    Lagnese, J., Leugering, G., Schmidt, E.: On the analysis and control of hyperbolic systems associated with vibrating networks. Proc. R. Soc. Edinb. Sect. A 124(1), 77–104 (1994)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Tikhonov, A.N., Samarskii, A.A.: Equations of Mathematical Physics. Courier Corporation, North Chelmsford (2013)zbMATHGoogle Scholar
  16. 16.
    Zuazua, E.: Control and stabilization of waves on 1-d networks. In: Aswini, A. (ed.) Modelling and Optimisation of Flows on Networks. Lecture Notes in Mathematics, vol. 2062, pp. 463–493. Springer, Heidelberg (2013)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsUniversity of Alaska FairbanksFairbanksUSA

Personalised recommendations