Chinese Annals of Mathematics, Series B

, Volume 38, Issue 2, pp 473–488 | Cite as

Exact boundary controllability for a coupled system of wave equations with Neumann boundary controls



This paper first shows the exact boundary controllability for a coupled system of wave equations with Neumann boundary controls. In order to establish the corresponding observability inequality, the authors introduce a compact perturbation method which does not depend on the Riesz basis property, but depends only on the continuity of projection with respect to a weaker norm, which is obviously true in many cases of application. Next, in the case of fewer Neumann boundary controls, the non-exact boundary controllability for the initial data with the same level of energy is shown.


Compactness-uniqueness perturbation Boundary observability Exact boundary controllability Non-exact boundary controllability Coupled system of wave equations Neumann boundary condition 

2000 MR Subject Classification

93B05 93B07 93C20 35L53 


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Part of the work was done during the visit of the second author at the Laboratoire International Associé Sino-Français de Mathématiques Appliquées (LIASFMA) and the School of Mathematical Sciences of Fudan University during June–August 2014. He would like to thank their hospitality and support.

The authors are very grateful to Professor Xu Zhang for bringing their attention to the references [11, 12] and for the valuable discussions on several occasions, and would like also to thank the referees for their very valuable comments and remarks, which were greatly appreciated to improve the presentation of the paper.


  1. [1]
    Alabau-Boussouira, F., A two-level energy method for indirect boundary observability and controllability of weakly coupled hyperbolic systems, SIAM J. Control Optim., 42, 2003, 871–904.MathSciNetCrossRefMATHGoogle Scholar
  2. [2]
    Alabau-Boussouira, F., A hierarchic multi-level energy method for the control of bidiagonal and mixed ncoupled cascade systems of PDEs by a reduced number of controls, Adv. Diff. Equ., 18, 2013, 1005–1072.MATHGoogle Scholar
  3. [3]
    Ammar Khodja, F., Benabdallah, A., Gonzalez-Burgos, M. and de Teresa, L., Recent results on the controllability of linear coupled parabolic problems: A survey, Math. Control Relat. Fields, 1, 2011, 267–306.MathSciNetCrossRefMATHGoogle Scholar
  4. [4]
    Bardos, C., Lebeau, G. and Rauch, J., Sharp sufficient conditions for the observation, control, and stabilization of waves from the boundary, SIAM J. Control Optim., 30, 1992, 1024–1064.CrossRefMATHGoogle Scholar
  5. [5]
    Brezis, H., Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer-Verlag, New York, 2011.MATHGoogle Scholar
  6. [6]
    Dehman, B., Le Rousseau, J. and Léautaud, M., Controllability of two coupled wave equations on a compact manifold, Arch. Ration. Mech. Anal., 211, 2014, 113–187.MathSciNetCrossRefMATHGoogle Scholar
  7. [7]
    Duyckaerts, T., Zhang, X. and Zuazua, E., On the optimality of the observability inequalities for parabolic and hyperbolic systems with potentials, Ann. Inst. H. Poincaré Anal. Non Linéaire, 25, 2008, 1–41.MathSciNetCrossRefMATHGoogle Scholar
  8. [8]
    Hu, L., Ji, F. Q. and Wang, K., Exact boundary controllability and exact boundary observability for a coupled system of quasilinear wave equations, Chin. Ann. Math., Ser. B, 34(4), 2013, 479–490.MathSciNetCrossRefMATHGoogle Scholar
  9. [9]
    Komornik, V., Exact Controllability and Stabilization, The Multiplier Method, Masson, Paris, 1994.MATHGoogle Scholar
  10. [10]
    Komornik, V. and Loreti, P., Observability of compactly perturbed systems, J. Math. Anal. Appl., 243, 2000, 409–428.MathSciNetCrossRefMATHGoogle Scholar
  11. [11]
    Lasiecka, I. and Triggiani, R., Trace regularity of the solutions of the wave equation with homogeneous Neumann boundary conditions and data supported away from the boundary, J. Math. Anal. Appl., 141, 1989, 49–71.MathSciNetCrossRefMATHGoogle Scholar
  12. [12]
    Lasiecka, I. and Triggiani, R., Sharp regularity for mixed second-order hyperbolic equations of Neumann type, I. L2 nonhomogeneous data, Ann. Mat. Pura Appl., 157, 1990, 285–367.CrossRefMATHGoogle Scholar
  13. [13]
    Lasiecka, I., Triggiani, R. and Zhang, X., Nonconservative wave equations with unobserved Neumann B. C.: Global uniqueness and observability in one shot, Differential geometric methods in the control of partial differential equations, Contemp. Math., 268, A. M. S., Providence, RI,2000, 227–325.MATHGoogle Scholar
  14. [14]
    Li, T.-T. and Rao, B. P., Exact synchronization for a coupled system of wave equations with Dirichlet boundary controls, Chin. Ann. Math., Ser. B, 34(1), 2013, 139–160.MathSciNetCrossRefMATHGoogle Scholar
  15. [15]
    Li, T.-T. and Rao, B. P., Asymptotic controllability and asymptotic synchronization for a coupled system of wave equations with Dirichlet boundary controls, Asym. Anal., 86, 2014, 199–226.MathSciNetMATHGoogle Scholar
  16. [16]
    Li, T.-T. and Rao, B. P., A note on the exact synchronization by groups for a coupled system of wave equations, Math. Methods Appl. Sci., 38, 2015, 241–246.MathSciNetCrossRefMATHGoogle Scholar
  17. [17]
    Li, T.-T. and Rao, B. P., Criteria of Kalman’s type to the approximate controllability and the approximate synchronization for a coupled system of wave equations with Dirichlet boundary controls, C. R. Acad. Sci. Paris, Ser. I, 353, 2015, 63–68.Google Scholar
  18. [18]
    Lions, J.-L. and Magenes, E., Problèmes aux Limites Non Homogènes et Applications, Vol. 1, Dunod, Paris, 1968.MATHGoogle Scholar
  19. [19]
    Lions, J.-L., Exact controllability, stabilization and perturbations for distributed systems, SIAM Rev., 30, 1988, 1–68.MATHGoogle Scholar
  20. [20]
    Lions, J.-L., Controlabilité Exacte, Perturbations et Stabilisation de Syst mes Distribués, Vol. 1, Masson, Paris, 1988.Google Scholar
  21. [21]
    Liu, Z. and Rao, B. P., A spectral approach to the indirect boundary control of a system of weakly coupled wave equations, Discrete Contin. Dyn. Syst., 23, 2009, 399–414.MathSciNetCrossRefMATHGoogle Scholar
  22. [22]
    Mehrenberger, M., Observability of coupled systems, Acta Math. Hungar., 103, 2004, 321–348.MathSciNetCrossRefMATHGoogle Scholar
  23. [23]
    Pazy, A., Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York, 1983.CrossRefMATHGoogle Scholar
  24. [24]
    Rosier, L. and de Teresa, L., Exact controllability of a cascade system of conservative equations, C. R. Math. Acad. Sci., Paris, 349, 2011, 291–296.MATHGoogle Scholar
  25. [25]
    Russell, D. L., Controllability and stabilization theory for linear partial differential equations: Recent progress and open questions, SIAM Rev., 20, 1978, 639–739.MathSciNetCrossRefMATHGoogle Scholar
  26. [26]
    Yao, P. F., On the observability inequalities for exact controllability of wave equations with variable coefficients, SIAM J. Control Optim., 37, 1999, 1568–1599.MathSciNetCrossRefMATHGoogle Scholar
  27. [27]
    Zhang, X. and Zuazua, E., A sharp observability inequality for Kirchhoff plate systems with potentials, Comput. Appl. Math., 25, 2006, 353–373.MathSciNetMATHGoogle Scholar

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© Fudan University and Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  1. 1.School of Mathematical SciencesFudan UniversityShanghaiChina
  2. 2.Shanghai Key Laboratory for Contemporary Applied MathematicsFudan UniversityShanghaiChina
  3. 3.Nonlinear Mathematical Modeling and Methods LaboratoryFudan UniversityShanghaiChina
  4. 4.Institut de Recherche Mathématique AvancéeUniversité de StrasbourgStrasbourgFrance

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