On the Exact Boundary Control for the Linear Klein-Gordon Equation in Non-cylindrical Domains

R. S. O. Nunes


The purpose of this paper is to study an exact boundary controllability problem in noncylindrical domains for the linear Klein-Gordon equation. Here, we work near of the extension techniques presented By J. Lagnese in [12] which is based in the Russell’s controllability method. The control time is obtained in any time greater then the value of the diameter of the domain on which the initial data are supported. The control is square integrable and acts on whole boundary and it is given by conormal derivative associated with the above-referenced wave operator.


Exact Boundary Controllability; Non-cylindrical Domains; Linear Klein-Gordon Equation.

Full Text:



C. Bardos and G. Chen, “Control and stabilization for wave equation, part iii: domain with moving boundary,” SIAM J. Control Optim., vol. 19, pp. 123–138,1981.

J. Lagnese, “On the support of solutions of the wave equation with applications to exact boundary value controllability,” J. Math. pures et appl., vol. 58, p. 121135, 1979.

L. Cui, X. Liu, and H. Gao, “Exact controllability for a one-dimensional waveequation in non-cylindrical domains,” J. Math. Anal. Appl., vol. 402, pp. 612–625, 2013.

M. M. Miranda, “Exact controllability for the wave equation in domains with variable boundary,” Rev. Mat. Univ. Complut. Madrid, vol. 9, pp. 435–457,1996.

W. D. Bastos and J. Ferreira, “Exact boundary control for the wave equation in a polyhedral time-dependent domain.,” App. Math. Lett., vol. 12, pp. 1–5,1999.

R. Dautray and J.-L. Lions, Mathematical Analysis and Numerical Methods for Science and Technology, Vol. 2. Berlin: Cambridge University Press, 1988.

F. G. Friedlander, Sound Pulses. Cambridge: Cambridge University Press,1958.

D. L. Russell, “A unified boundary controllability theory for hyperbolic and parabolic partial differential equations,” Stud. Appl. Math., vol. 52, pp. 189–211, 1973.

D. Tataru, “On regularity of the boundary traces for the wave,” Ann. Scuola Norm. Pisa, C. L. Sci., vol. 26, pp. 185–206, 1998.

R. S. O. Nunes and W. D. Bastos, “Energy decay for the linear klein-gordon equation and boundary control,” J. Math. Anal. Appl., vol. 414, pp. 934–944, 2014.

R. S. O. Nunes and W. D. Bastos, “Analyticity and near optimal time boundary controllability for the linear klein-gordon equation,” J. Math. Anal. Appl., vol. 445, pp. 394–406, 2017.

T. Kato, Perturbation theory for linear operators. New York: Springer-Verlag,1966.

DOI: https://doi.org/10.5540/tema.2020.021.02.371

Article Metrics

Metrics Loading ...

Metrics powered by PLOS ALM


  • There are currently no refbacks.

Trends in Computational and Applied Mathematics

A publication of the Brazilian Society of Applied and Computational Mathematics (SBMAC)


Indexed in:



Desenvolvido por:

Logomarca da Lepidus Tecnologia