Suboptimal boundary controls for elliptic equation in critically perforated domain
Annales de l'I.H.P. Analyse non linéaire, Volume 25 (2008) no. 6, p. 1073-1101
@article{AIHPC_2008__25_6_1073_0,
     author = {D'Apice, Ciro and De Maio, Umberto and Kogut, Peter I.},
     title = {Suboptimal boundary controls for elliptic equation in critically perforated domain},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     publisher = {Elsevier},
     volume = {25},
     number = {6},
     year = {2008},
     pages = {1073-1101},
     doi = {10.1016/j.anihpc.2007.07.001},
     zbl = {1170.35015},
     mrnumber = {2466322},
     language = {en},
     url = {http://www.numdam.org/item/AIHPC_2008__25_6_1073_0}
}
D'Apice, Ciro; De Maio, Umberto; Kogut, Peter I. Suboptimal boundary controls for elliptic equation in critically perforated domain. Annales de l'I.H.P. Analyse non linéaire, Volume 25 (2008) no. 6, pp. 1073-1101. doi : 10.1016/j.anihpc.2007.07.001. http://www.numdam.org/item/AIHPC_2008__25_6_1073_0/

[1] Bouchitté G., Fragala I., Homogenization of thin structures by two-scale method with respect to measures, SIAM J. Math. Anal. 32 (6) (2001) 1198-1226. | MR 1856245 | Zbl 0986.35015

[2] G. Buttazzo, Γ-convergence and its applications to some problem in the calculus of variations, in: School on Homogenization, ICTP, Trieste, September 6-17, 1993, 1994, pp. 38-61.

[3] Buttazzo G., Dal Maso G., Γ-convergence and optimal control problems, J. Optim. Theory Appl. 32 (1982) 385-407. | MR 686213 | Zbl 0471.49012

[4] Carbone L., De Arcangelis R., Unbounded Functionals in the Calculus of Variations. Representation, Relaxation, and Homogenization, Chapman and Hall/CRC, New York, 2002. | MR 1910459 | Zbl 1002.49018

[5] Cardone G., D'Apice C., De Maio U., Homogenization in perforated domains with mixed conditions, Nonlinear Diff. Equ. Appl. 9 (2002) 246-325. | MR 1917377 | Zbl 1046.35007

[6] Casado-Díaz J., Existence of a sequence satisfying Cioranescu-Murat conditions in homogenization of Dirichlet problems in perforated domains, Rend. Mat. Appl. (7) 16 (1996) 387-413. | MR 1422390 | Zbl 0870.35013

[7] Cioranescu D., Donato P., Murat F., Zuazua E., Homogenization and correctors for the wave equation in domains with small holes, Ann. Sc. Norm. Super. Pisa, Sc. Fis. Mat. 17 (4) (1991) 251-293. | Numdam | MR 1129303 | Zbl 0807.35077

[8] Cioranescu D., Donato P., Zuazua E., Exact boundary controllability for the wave equation in domains with small holes, J. Math. Pures Appl. 71 (1992) 343-377. | MR 1176016 | Zbl 0843.35009

[9] Cioranescu D., Murat F., Un terme étrage venu d'ailleurs, in: Nonlinear Partial Differential Equations and their applications. Collége de France Seminar, Research Notes in Mathematics, Pitman, London, 1981, vol. II, pp. 58-138, vol. III, pp. 157-178. | Zbl 0498.35034

[10] Cioranescu D., Saint Jean Paulin J., Homogenization in open sets with holes, J. Math. Anal. Appl. 71 (1978) 590-607. | MR 548785 | Zbl 0427.35073

[11] Conca C., Donato P., Nonhomogeneous Neumann problems in domains with small holes, Modélisation Mathématique et Analyse Numérique 22 (4) (1988) 561-608. | Numdam | MR 974289 | Zbl 0669.35028

[12] Corbo Esposito A., D'Apice C., Gaudiello A., A homogenization problem in a perforated domain with both Dirichlet and Neumann conditions on the boundary of the holes, Asymptodic Anal. 31 (2002) 297-316. | MR 1937842 | Zbl 1043.35027

[13] Coron J.-M., Crépeau E., Exact boundary controllability of a nonlinear KdV equation with critical length, J. Eur. Math. Soc. (JEMS) 6 (3) (2004) 367-398. | MR 2060480 | Zbl 1061.93054

[14] Dal Maso G., Murat F., Asymptotic behaviour and correctors for Dirichlet problem in perforated domains with homogeneous monotone operators, Ann. Sc. Norm. Sup. Pisa Cl. Sci. 24 (4) (1997) 239-290. | Numdam | MR 1487956 | Zbl 0899.35007

[15] Evans L.C., Gariepy R.F., Measure Theory and Fine Properties of Functions, CRC Press, Boca Raton, FL, 1992. | MR 1158660 | Zbl 0804.28001

[16] Fursikov A.V., Optimal Control of Distributed Systems. Theory and Applications, Amer. Math. Soc., 2000. | MR 1726442 | Zbl 1027.93500

[17] Ioffe A.D., Tikhomirov V.M., Theory of Extremal Problems, Nauka, Moskow, 1974, (in Russian). | MR 410502

[18] Kesavan S., Saint Jean Paulin J., Optimal control on perforated domains, J. Math. Anal. Appl. 229 (1999) 563-586. | MR 1666365 | Zbl 0919.49005

[19] Kogut P.I., S-convergence in homogenization theory of optimal control problems, Ukrain. Mat. Zh. 49 (11) (1997) 1488-1498, (in Russian); English transl. in:, Ukrainian Math. J. 49 (11) (1997) 1671-1682. | MR 1672876 | Zbl 0933.93026

[20] Kogut P.I., Leugering G., On S-homogenization of an optimal control problem with control and state constraints, J. Anal. Appl. 20 (2) (2001) 395-429. | MR 1846609 | Zbl 0982.35014

[21] Lions J.L., Équations différentielles opérationnelles, Springer-Verlag, Berlin, 1961. | MR 153974 | Zbl 0098.31101

[22] Lions J.L., Optimal Control of Systems Governed by Partial Differential Equations, Springer-Verlag, New York, 1971. | MR 271512 | Zbl 0203.09001

[23] Marchenko V.A., Khruslov E.Ya., Boundary Value Problems in Domain with Fine-Grained Boundary, Naukova Dumka, Kyiv, 1974. | Zbl 0289.35002

[24] Nandakumar A.K., Convergence of the boundary control for the wave equation in domains with holes of critical size, Electron. J. Differential Equations 2002 (35) (2002) 1-10. | MR 1907711 | Zbl 1007.35007

[25] Pastukhova S.E., On the convergence of hyperbolic semigroups in variable Hilbert spaces, J. Math. Sci. 127 (5) (2005) 2263-2283. | MR 2360842 | Zbl 1126.47038

[26] Saint Jean Paulin J., Zoubairi H., Optimal control and “strange term” for the Stokes problem in perforated domains, Portugal. Math. 59 (2) (2002) 161-178. | MR 1907412 | Zbl 1017.49005

[27] Scrypnik I.V., Averaging nonlinear Dirichlet problems in domains with channels, Soviet Math. Dokl. 42 (1991) 853-857. | MR 1100827 | Zbl 0757.35027

[28] Zhikov V.V., On an extension of the method of two-scale convergence and its applications, Sbornik Math. 191 (7) (2000) 973-1014. | MR 1809928 | Zbl 0969.35048

[29] Zhikov V.V., Kozlov S.M., A Oleinik O., Homogenization of Differential Operators and Integral Functionals, Springer-Verlag, Berlin, 1994. | MR 1329546 | Zbl 0838.35001