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

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Abstract

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.

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References

  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.

  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.

  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.

  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.

  5. [5]

    Brezis, H., Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer-Verlag, New York, 2011.

  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.

  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.

  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.

  9. [9]

    Komornik, V., Exact Controllability and Stabilization, The Multiplier Method, Masson, Paris, 1994.

  10. [10]

    Komornik, V. and Loreti, P., Observability of compactly perturbed systems, J. Math. Anal. Appl., 243, 2000, 409–428.

  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.

  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.

  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.

  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.

  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.

  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.

  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.

  18. [18]

    Lions, J.-L. and Magenes, E., Problèmes aux Limites Non Homogènes et Applications, Vol. 1, Dunod, Paris, 1968.

  19. [19]

    Lions, J.-L., Exact controllability, stabilization and perturbations for distributed systems, SIAM Rev., 30, 1988, 1–68.

  20. [20]

    Lions, J.-L., Controlabilité Exacte, Perturbations et Stabilisation de Syst mes Distribués, Vol. 1, Masson, Paris, 1988.

  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.

  22. [22]

    Mehrenberger, M., Observability of coupled systems, Acta Math. Hungar., 103, 2004, 321–348.

  23. [23]

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

  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.

  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.

  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.

  27. [27]

    Zhang, X. and Zuazua, E., A sharp observability inequality for Kirchhoff plate systems with potentials, Comput. Appl. Math., 25, 2006, 353–373.

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Acknowledgments

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.

Author information

Correspondence to Tatsien Li.

Additional information

Dedicated to Professor Haim Brezis on the occasion of his 70th birthday

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

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Li, T., Rao, B. Exact boundary controllability for a coupled system of wave equations with Neumann boundary controls. Chin. Ann. Math. Ser. B 38, 473–488 (2017). https://doi.org/10.1007/s11401-017-1078-5

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Keywords

  • 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