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Exact Boundary Controllability and Energy Decay for a System of Wave Equations Linearly Coupled

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Abstract

The aim of this paper is to study an exact boundary controllability problem and local energy decay for a system of m linearly coupled wave equations . We obtain control time near of the optimal time with control of Neuman type acting on all boundary of the considered domain. We also show that the local energy decays at a rate \(t^{-n}\), where n is the spatial dimension of the domain where the initial data are supported.

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References

  1. Alabau-Boussouira, F.: A two-level energy method for indirect boundary observability and controllability of weak coupled hyperbolic systems SIAM. J. Control Optm. 42(3), 871–906 (2003)

    Article  Google Scholar 

  2. Bardos, C., Lebeau, G., Rauch, J.: Sharp sufficient conditions for the observation, control, and stabilization of waves from the boundary. SIAM J. Control Optim. 30(5), 1024–1065 (1992)

    Article  MathSciNet  Google Scholar 

  3. Russell, D.L.: A unified boundary controllability theory for hyperbolic and parabolic partial differential equations Stud. Appl. Math. 52, 189–211 (1973)

    MathSciNet  MATH  Google Scholar 

  4. Tataru, D.: On regularity of the boundary traces for the wave. Ann. Scuola Norm. Pisa C. L. Sci.(4) 26(1), 185–206 (1998)

    MathSciNet  MATH  Google Scholar 

  5. Atkinson, F.V.: A spectral problem for completely continuous operators. Acta Math. Acad. Sci. Hungar 3, 53–60 (1952)

    Article  MathSciNet  Google Scholar 

  6. Watson, G.N.: A treatise on the theory of Bessel Function, 2nd edn. Cambridge University Press, London (1966)

    MATH  Google Scholar 

  7. Folland, G.B.: Fourier analysis and its applications. Wadsworth and Brooks, California (1992)

    MATH  Google Scholar 

  8. Lagnese, J.: Boundary value control of a class of hyperbolic equations in a general region. SIAM J. Control Optim. 15(6), 973–983 (1977)

    Article  MathSciNet  Google Scholar 

  9. Lions, J.L.: Exact controllability, stabilization and perturbations for distributed system. SIAM Rev. 30(1), 1–68 (1988)

    Article  MathSciNet  Google Scholar 

  10. Wu, J., Li, S., Chai, S.: Exact controllability of wave equations with variable coefficients coupled in parallel Asian. J. Control 12(5), 650–655 (2010)

    MathSciNet  Google Scholar 

  11. Rajaram, R., Najafi, M.: Exact controllability of wave equations in \(R^{n}\) coupled in parallel. J. Math. Anal. Appl. 356, 7–12 (2009)

    Article  MathSciNet  Google Scholar 

  12. Nunes, R.S.O., Bastos, W.D.: Analyticity and near optimal time boundary controllability for the linear Klein–Gordon equation. J. Math. Anal. Appl. 445, 394–406 (2017)

    Article  MathSciNet  Google Scholar 

  13. Nunes, R.S.O., Bastos, W.D., Pitot, J.M.S.: Energy decay and control for a system of strings elastically connected in parallel. Math. Methods Appl. Sci. 43, 1230–1242 (2020)

    Article  MathSciNet  Google Scholar 

  14. Nunes, R.S.O., Bastos, W.D.: Energy decay for the linear Klein–Gordon equation and boundary control. J. Math. Anal. Appl. 414, 934–944 (2014)

    Article  MathSciNet  Google Scholar 

  15. Kelly, S.G.: Free and forced vibrations of elastically connected structures. Adv. Acoust. Vib. 2010(984361), 11 (2010)

  16. Kelly, S.G., Nicely, C.: Free vibrations of a series of beams connected by viscoelastic layers. Adva. Acoust. Vib. 2015(976841), 8 (2015)

  17. Kato, T.: Perturbation theory for linear operators. Springer, New York (1966)

    Book  Google Scholar 

  18. Bastos, W.D., Spezamiglio, A., Raposo, C.A.: On exact boundary controllability for linearly coupled wave equations. J. Math. Anal. Appl. 381, 557–564 (2011)

    Article  MathSciNet  Google Scholar 

  19. Bastos, W.D., Nunes, R.S.O., Pitot, J.M.S.: Exact boundary controllability for a series of membranes elastically connected. Electron. J. Differ. Equ. 06, 1–16 (2017)

    MathSciNet  MATH  Google Scholar 

  20. Oniszczuk, Z.: Transverse vibrations of elastically connected double-string complex system, part I: free vibration. J. Sound Vib. 232(2), 355–366 (2000)

    Article  Google Scholar 

  21. Oniszczuk, Z.: Transverse vibrations of elastically connected rectangular double-membrane compound system. J. Sound Vib. 221(2), 235–250 (1999)

    Article  Google Scholar 

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  1. Federal University of Mato Grosso, Av. Fernando Corrêa da Costa, 2367, Cuiabá, MT, 78060-900, Brazil

    Ruikson S. O. Nunes

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Nunes, R.S.O. Exact Boundary Controllability and Energy Decay for a System of Wave Equations Linearly Coupled. Mediterr. J. Math. 18, 30 (2021). https://doi.org/10.1007/s00009-020-01672-7

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  • DOI: https://doi.org/10.1007/s00009-020-01672-7

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