Suboptimal boundary controls for elliptic equation in critically perforated domain

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1. [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. Zbl0986.35015MR1856245
2. [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. [3] Buttazzo G., Dal Maso G., Γ-convergence and optimal control problems, J. Optim. Theory Appl.32 (1982) 385-407. Zbl0471.49012MR686213
4. [4] Carbone L., De Arcangelis R., Unbounded Functionals in the Calculus of Variations. Representation, Relaxation, and Homogenization, Chapman and Hall/CRC, New York, 2002. Zbl1002.49018MR1910459
5. [5] Cardone G., D'Apice C., De Maio U., Homogenization in perforated domains with mixed conditions, Nonlinear Diff. Equ. Appl.9 (2002) 246-325. Zbl1046.35007MR1917377
6. [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. Zbl0870.35013MR1422390
7. [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. Zbl0807.35077MR1129303
8. [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. Zbl0843.35009MR1176016
9. [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. Zbl0498.35034
10. [10] Cioranescu D., Saint Jean Paulin J., Homogenization in open sets with holes, J. Math. Anal. Appl.71 (1978) 590-607. Zbl0427.35073MR548785
11. [11] Conca C., Donato P., Nonhomogeneous Neumann problems in domains with small holes, Modélisation Mathématique et Analyse Numérique22 (4) (1988) 561-608. Zbl0669.35028MR974289
12. [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. Zbl1043.35027MR1937842
13. [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. Zbl1061.93054MR2060480
14. [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. Zbl0899.35007MR1487956
15. [15] Evans L.C., Gariepy R.F., Measure Theory and Fine Properties of Functions, CRC Press, Boca Raton, FL, 1992. Zbl0804.28001MR1158660
16. [16] Fursikov A.V., Optimal Control of Distributed Systems. Theory and Applications, Amer. Math. Soc., 2000. Zbl1027.93500MR1726442
17. [17] Ioffe A.D., Tikhomirov V.M., Theory of Extremal Problems, Nauka, Moskow, 1974, (in Russian). MR410502
18. [18] Kesavan S., Saint Jean Paulin J., Optimal control on perforated domains, J. Math. Anal. Appl.229 (1999) 563-586. Zbl0919.49005MR1666365
19. [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. Zbl0933.93026MR1672876
20. [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. Zbl0982.35014MR1846609
21. [21] Lions J.L., Équations différentielles opérationnelles, Springer-Verlag, Berlin, 1961. Zbl0098.31101MR153974
22. [22] Lions J.L., Optimal Control of Systems Governed by Partial Differential Equations, Springer-Verlag, New York, 1971. Zbl0203.09001MR271512
23. [23] Marchenko V.A., Khruslov E.Ya., Boundary Value Problems in Domain with Fine-Grained Boundary, Naukova Dumka, Kyiv, 1974. Zbl0289.35002
24. [24] Nandakumar A.K., Convergence of the boundary control for the wave equation in domains with holes of critical size, Electron. J. Differential Equations2002 (35) (2002) 1-10. Zbl1007.35007MR1907711
25. [25] Pastukhova S.E., On the convergence of hyperbolic semigroups in variable Hilbert spaces, J. Math. Sci.127 (5) (2005) 2263-2283. Zbl1126.47038MR2360842
26. [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. Zbl1017.49005MR1907412
27. [27] Scrypnik I.V., Averaging nonlinear Dirichlet problems in domains with channels, Soviet Math. Dokl.42 (1991) 853-857. Zbl0757.35027MR1100827
28. [28] Zhikov V.V., On an extension of the method of two-scale convergence and its applications, Sbornik Math.191 (7) (2000) 973-1014. Zbl0969.35048MR1809928
29. [29] Zhikov V.V., Kozlov S.M., A Oleinik O., Homogenization of Differential Operators and Integral Functionals, Springer-Verlag, Berlin, 1994. Zbl0838.35001MR1329546