Abstract
The stabilization of the wave equation in a polygonal domain with cracks is analyzed. Using the multiplier method, we show that a boundary stabilization augmented by an internal one concentrated in a small neighbourhood of the cracks lead to the exponential stability of the problem.
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Adams, R.: Sobolev Spaces. Academic Press, New York (1994)
Brossard, R., Lohéac, J.-P.: Stabilisation frontiére du systéme élastodynamique dans un polygone plan. C. R. Acad. Sci. Paris, Série I. 338, 213–218 (2004)
Brossard, R., Lohéac, J.-P.: Boundary stabilization of elastodynamic systems, II. The case of a linear feedback. J. Dyn. Control Syst. 16, 355–375 (2010)
Cornilleau, P., Lohéac, J.-P., Osses, A.: Nonlinear Neumann boundary stabilization of the wave equation using rotated multipliers. J. Dyn. Control Syst. 16, 163–188 (2010)
Grisvard, P.: Singularities in Boundary Value Problems, RMA 22. Springer, Berlin (1992)
Grisvard, P.: Contrôlabilité exacte de l’équation des ondes en présence de singularités. J. Math. Pures Appl. 68, 215–259 (1989)
Grisvard, P.: Contrôlabilité exacte dans les polygones et polyèdres. C. R. Acad. Sci. Paris Ser. I(304), 367–370 (1987)
Grisvard, P.: Contrôlabilité exacte avec des conditions mêlées. C. R. Acad. Sci. Paris Ser. I(305), 363–366 (1987)
Grisvard, P.: Elliptic problems in non smooth domains. In: Monograph Studies in Mathematics, vol. 24. Pitman, London (1985)
Komornik, V.: Exact controlabillity and stabilization–The multiplier method. Research in Applied Mathematics. Masson, Paris (1994)
Lions, J.-L.: Contrôlabilité exacte, perturbation des systèmes distribués, Tome 1. Masson, Paris (1988)
Liu, K.: Locally distributed control and damping for the conservative systems. SIAM J. Control Optim. 35, 1574–1590 (1997)
Martinez, P.: Boundary stabilization of the wave equation in almost star-shaped domain. SIAM J. Control Optim. 37, 673–694 (1999)
Martinez, P.: A new method to obtain decay rate estimates for dissipative systems with localized damping. Rev. Mat. Complut. 12, 251–283 (1999)
Martinez, P.: Stabilisation frontière de l’équation des ondes dans des domaines polygonaux. C. R. Acad. Sci. Paris Ser. I(322), 365–370 (1996)
Moussaoui, M.: Singularités des solutions du problème mêlé, contrôlabilité exacte et stabilisation frontière. ESAIM Proc. 2, 195–201 (1997)
Niane, M.T., Seck, O.: Contrôlabilité exacte frontìere de l’équation des ondes en présence de fissures par adjonction de contrôles internes au voisinage des sommets de fissures. C. R Acad. Sci. Paris Ser. I(316), 695–700 (1993)
Nicaise, S.: Boundary exact controlability of interface problems with singularities I: addition of the coefficients of singularities. SIAM J. Control Optim. 34, 1512–1532 (1996)
Nicaise, S.: Boundary exact controlability of interface problems with singularities II: Addition of internal controls. SIAM J. Control Optim. 35, 585–603 (1997)
Zuazua, E.: Exponential decay for the semilinear wave equation with locally distributed damping. Commun. Partial Diff. Equ. 15, 205–235 (1990)
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Bayili, G., Nicaise, S. Stabilization of the wave equation in a polygonal domain with cracks. Rev Mat Complut 27, 259–289 (2014). https://doi.org/10.1007/s13163-012-0113-z
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DOI: https://doi.org/10.1007/s13163-012-0113-z