Skip to main content
SpringerLink
Log in

Exact Boundary Synchronization for a Coupled System of Wave Equations with Neumann Boundary Controls

  • Published:
Chinese Annals of Mathematics, Series B Aims and scope Submit manuscript

Abstract

In this paper, for a coupled system of wave equations with Neumann boundary controls, the exact boundary synchronization is taken into consideration. Results are then extended to the case of synchronization by groups. Moreover, the determination of the state of synchronization by groups is discussed with details for the synchronization and for the synchronization by 3-groups, respectively.

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

Similar content being viewed by others

References

  1. Bardos, C., Lebeau, G. and Rauch, J., Sharp sufficient conditions for the observation, control, and stabi- lization of waves from the boundary, SIAM J. Control Optim., 30, 1992, 1024–1065.

    Article  MathSciNet  MATH  Google Scholar 

  2. Hu, L., Li, T.-T. and Rao, B. P., Exact boundary synchronization for a coupled system of 1-D wave equations with coupled boundary conditions of dissipative type, Communications on Pure and Applied Analysis, 13, 2014, 881–901.

    Article  MathSciNet  MATH  Google Scholar 

  3. Huygens, C., Horologium Oscillatorium Sive de Motu Pendulorum ad Horologia Aptato Demonstrationes Geometricae, Apud F. Muguet, Parisiis, 1673.

    Google Scholar 

  4. Li, T.-T. and Rao, B. P., Exaxt synchronization for a coupled system of wave equation with Dirichlet boundary controls, Chin. Ann. Math., Ser. B, 34(1), 2013, 139–160.

    Article  MathSciNet  MATH  Google Scholar 

  5. Li, T.-T. and Rao, B. P., A note on the exact synchronization by groups for a coupled system of wave equations, Math. Meth. Appl. Sci., 38(13), 2015, 2803–2808.

    Article  MathSciNet  MATH  Google Scholar 

  6. Li, T.-T. and Rao, B. P., On the exactly synchronizable state to a coupled system of wave equations, Portugaliae Math., 72, 2015, 83–100.

    Article  MathSciNet  MATH  Google Scholar 

  7. Li, T.-T. and Rao, B. P., Exact synchronization by groups for a coupled system of wave equations with Dirichlet boundary controls, J. Math. Pures Appl., 105(1), 2016, 86–101.

    Article  MathSciNet  MATH  Google Scholar 

  8. Li, T.-T. and Rao, B. P., Exact boundary controllability for a coupled system of wave equations with Neumann controls, Chin. Ann. Math., Ser. B, 38(2), 2017, 473–488.

    Article  MathSciNet  MATH  Google Scholar 

  9. Li, T.-T., Rao, B. P. and Hu, L., Exact boundary synchronization for a coupled system of 1-D wave equations, ESAIM: Control, Optimisation and Calculus of Variations, 20, 2014, 339–361.

    Article  MathSciNet  MATH  Google Scholar 

  10. Li, T.-T., Rao, B. P. and Wei, Y. M., Generalized exact boundary synchronization for a coupled system of wave equations, Discrete Contin. Dyn. Syst., 34, 2014, 2893–2905.

    Article  MathSciNet  MATH  Google Scholar 

  11. Lions, J.-L. and Magenes, E., Non-homogeneous Boundary Value Problems and Applications, I, Springer-Verlag, New York, Heidelberg, 1972.

    Book  MATH  Google Scholar 

  12. Pazy, A., Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences, 44, Springer-Verlag, New York, 1983.

    Google Scholar 

  13. Wiener, N., Cybernetics, or Control and Communication in the Animal and the Machine, 2nd ed., the M.I.T. Press, Cambridge, Mass., John Wiley & Sons, Inc., New York, London, 1961.

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Mathematical Sciences, Fudan University, Shanghai, 200433, China

    Tatsien Li & Xing Lu

  2. Shanghai Key Laboratory for Contemporary Applied Mathematics, Shanghai, China

    Tatsien Li

  3. Nonlinear Mathematical Modeling and Methods Laboratory, Shanghai, China

    Tatsien Li

  4. Institut de Recherche Mathématique Avancée, Université de Strasbourg, 67084, Strasbourg, France

    Bopeng Rao

Authors
  1. Tatsien Li
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Xing Lu
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Bopeng Rao
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Tatsien Li.

Additional information

Dedicated to Philippe G. Ciarlet on the occasion of his 80th birthday

This work was supported by the National Natural Science Foundation of China (No. 11121101) and the National Basic Research Program of China (No. 2013CB834100).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Li, T., Lu, X. & Rao, B. Exact Boundary Synchronization for a Coupled System of Wave Equations with Neumann Boundary Controls. Chin. Ann. Math. Ser. B 39, 233–252 (2018). https://doi.org/10.1007/s11401-018-1062-8

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11401-018-1062-8

Keywords

2000 MR Subject Classification

Search

Navigation