Study and suppression of singularities in wave-type evolution equations on non-convex domains with cracks

One of the objectives of this paper is to establish the exact controllability for wave-type evolution equations on non-convex and/or cracked domains with non-concurrent support crack lines. Admittedly, we know that according to the work of Grisvard P., in domains with corners or cracks, the formulas of integrations by parts are subject to geometric conditions: the lines of cracks or their supports must be concurrent. In this paper, we have established the exact controllability for the wave equation in a domain with cracks without these additional geometric conditions.

Keywords:

Control, controllability, norms estimations, singularities cracks,

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Kondratiev, V.A., Boundary value problems for elliptic equation in domain with conical or angular points, Transactions Moscow Mat. Soc., (1967), 227-313.
Grisvard, P., Elliptic Problems in Nonsmooth Domains, Monographs and Studies in Mathematics, Vol. 24, Pitman (Advanced Publishing Program), Boston, MA, 1985.
Moussaoui, M., Singularites des solutions du probleme mele, controlabilite exacte et stabilisation frontiere, Soc. Math. Appl. Indust., 1997.
Niane, M.T., Bayili, G., Sene, A., Sene, A., Sy, M., Is it possible to cancel singularities in a domain with corners and cracks?, C. R. Math. Acad. Sci., 343(2) (2006), 115-118. https://doi:10.1016/j.crma.2006.05.003
Seck, C., Bayili, G., Sene, A., Niane, M.T., Controlabilite exacte de l'equation des ondes dans des espaces de Sobolev non reguliers pour un ouvert poygonal, Afrika Matematika, 23 (2012), 1-9. https://doi:10.1007/s13370-011-0001-6
Gilbert, B., Nicaise, S., Stabilization of the wave equation in a polygonal domain with cracks, Rev. Mat. Complut., 27(1) (2014), 259-289. https://doi: 10.1007/s13163-012-0113-z
Costabel, M., On the limit Sobolev regularity for Dirichlet and Neumann problems on Lipschitz domains, Math. Nachr., 292 (2019), 2165-2173. https://doi:10.1002/mana.201800077.
Niane, M.T., Controlabilite spectrale elargie des syst`emes distribues par une action sur une petite partie analytique arbitraire de la fontiere, C.R. Acad. Sci., Paris, 309(1), (1989), 335-340.
Lions, J.L., Controlabilite Exacte, Perturbations et Stabilisation de SystEMes DistribuES,Tome 2, Recherches en Mathematiques Appliquees, Research in Applied Mathematics, Volume 9, Perturbations, Masson, Paris, 1988.
Lions, J.-L., Controlabilite exacte, perturbations et stabilisation de systemes distribues. Tome 1, Recherches en Math´ematiques Appliquees ,Research in Applied Mathematics, Volume 8, Paris, 1988.
Brezis, H., Analyse Fonctionnelle, Theorie et Applications, Masson, 1983.
Hormander, L., Linear Partial Differential Operators, Springer Verlag, Berlin, 1976.
Dauge, M., Balac, S., Moitier, Z., Asymptotics for 2D whispering gallery modes in optical micro-disks with radially varying index, Arxiv: 2003.14315. https://doi:10.1093/imamat/hxab033.
Dauge, M., Costabel, M., Hu, J.Q., Characterization of Sobolev spaces by their Fourier coefficients in axisymmetric domains, arXiv:2004.07216v1, 2020.
Dauge, M., Initiation Into Corner Singularities, Course Given in the RICAM Special Semester on Computational Methods in Science and Engineering, October, 2016.
Costabel, M., On the limit Sobolev regularity for Dirichlet and Neumann problems on Lipschitz domains, Math. Nachr., 292 (2019), 2165-2173. arXiv: 1711.07179. https://doi:10.1002/mana.201800077