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

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Abstract

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.

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Notes

  1. In a tree graph there is only one edge between v and \(a_j\). The case where there are multiple edges between two vertices can be solved in the same way.

  2. Although \(v_n\) is a boundary vertex, we label it \(v_n\) instead of \(\gamma _n\), so we can refer to vertices on P as \(v_i\)’s together.

References

  1. Ali Mehmeti, F.: Nonlinear waves in networks. Math. Res. 80, 171 (1994)

    MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  3. Arioli, M., Benzi, M.: A finite element method for quantum graphs. IMA J. Numer. Anal. 38(3), 1119–1163 (2017)

    Article  MathSciNet  Google Scholar 

  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)

  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)

    MATH  Google Scholar 

  6. Avdonin, S., Kurasov, P.: Inverse problems for quantum trees. Inverse Probl. Imaging 2(1), 1–21 (2008)

    Article  MathSciNet  Google Scholar 

  7. Avdonin, S., Mikhaylov, V.: The boundary control approach to inverse spectral theory. Inverse Probl. 26(4), 045009 (2010)

    Article  MathSciNet  Google Scholar 

  8. Avdonin, S., Nicaise, S.: Source identification problems for the wave equation on graphs. Inverse Probl. 31(9), 095007 (2015)

    Article  MathSciNet  Google Scholar 

  9. Avdonin, S., Edward, J.: Exact controllability for string with attached masses. SIAM J. Control Optim. 56(2), 945–980 (2018)

    Article  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  12. Bergeron, F., Labelle, G., Leroux, P.: Combinatorial Species and Tree-Like Structures, vol. 67. Cambridge University Press, Cambridge (1998)

    MATH  Google Scholar 

  13. Dáger, R., Zuazua, E.: Wave Propagation, Observation and Control in 1-d Flexible Multi-structures, vol. 50. Springer, Berlin (2006)

    Book  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  15. Tikhonov, A.N., Samarskii, A.A.: Equations of Mathematical Physics. Courier Corporation, North Chelmsford (2013)

    Google Scholar 

  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)

    Chapter  Google Scholar 

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Acknowledgements

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

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Authors and Affiliations

  1. Department of Mathematics and Statistics, University of Alaska Fairbanks, Fairbanks, AK, 99775-6660, USA

    Sergei Avdonin & Yuanyuan Zhao

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  1. Sergei Avdonin
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  2. Yuanyuan Zhao
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Correspondence to Yuanyuan Zhao.

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The research of Sergei Avdonin was supported in part by the National Science Foundation, Grant DMS 1909869 and by the Ministry of Education and Science of Republic of Kazakhstan under the Grant No. AP05136197. The research of Yuanyuan Zhao was supported by the National Science Foundation Graduate Research Fellowship under Grant No. 1242789.

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Avdonin, S., Zhao, Y. Exact Controllability of the 1-D Wave Equation on Finite Metric Tree Graphs. Appl Math Optim 83, 2303–2326 (2021). https://doi.org/10.1007/s00245-019-09629-3

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  • DOI: https://doi.org/10.1007/s00245-019-09629-3

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