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Exact internal controllability for the wave equation in a domain with oscillating boundary with Neumann boundary condition
1. | Dipartimento di Matematica e Applicazioni, Università degli Studi di Napoli “Federico II”, DMA “R. Caccioppoli”, Complesso Monte S. Angelo, via Cintia, 80126 Napoli |
2. | Department of Mathematics, Indian Institute of Science, Bangalore-560012, India, India |
References:
[1] |
Y. Amirat and O. Bodart, Boundary layer correctors for the solution of laplace equation in a domain with oscillating boundary, Z. Anal. Anwendungen., 20 (2001), 929-940.
doi: 10.4171/ZAA/1052. |
[2] |
Y. Amirat, O. Bodart, U. De Maio and A. Gaudiello, Asymptotic approximation of the solution of the laplace equation in a domain with highly oscillating boundary, SIAM J. Math. Anal., 35 (2004), 1598-1616.
doi: 10.1137/S0036141003414877. |
[3] |
Y. Amirat, O. Bodart, U. De Maio and A. Gaudiello, Asymptotic approximation of the solution of Stokes equation in a domain with highly oscillating boundary, Ann. Univ. Ferrara, 53 (2007), 135-148.
doi: 10.1007/s11565-007-0015-z. |
[4] |
Y. Amirat, O. Bodart, U. De Maio and A. Gaudiello, Effective boundary condition for Stokes flow over a very rough surface, J. Differential Equations, 254 (2013), 3395-3430.
doi: 10.1016/j.jde.2013.01.024. |
[5] |
N. Ansini and A. Braides, Homogenization of oscillating boundaries and applications to thin films, J. Anal. Math., 83 (2001), 151-182.
doi: 10.1007/BF02790260. |
[6] |
V. Barbu and TH. Precupanu, Convexity and Optimization in Banach Spaces, D. Reidel, Dordrecht, 1986. |
[7] |
B. Birnir, S. Hou and N. Wellander, Derivation of the viscous Moore-Greitzer equation for aeroengine flow, J. Math. Phys., 48 (2007), 065209, 31pp.
doi: 10.1063/1.2534332. |
[8] |
D. Blanchard, L. Carbone and A. Gaudiello, Homogenization of a monotone problem in a domain with oscillating boundary, M2AN Math. Model. Numer. Anal., 33 (1999), 1057-1070.
doi: 10.1051/m2an:1999134. |
[9] |
D. Blanchard and A. Gaudiello, Homogenization of highly oscillating boundaries and reduction of dimension for a monotone problem, ESAIM Control Optim. Calc. Var., 9 (2003), 449-460.
doi: 10.1051/cocv:2003022. |
[10] |
D. Blanchard, A. Gaudiello and G. Griso, Junction of a periodic family of elastic rods with a 3d plate. Part I, J. Math. Pures Appl., 88 (2007), 1-33.
doi: 10.1016/j.matpur.2007.04.005. |
[11] |
D. Blanchard, A. Gaudiello and G. Griso, Junction of a periodic family of elastic rods with a thin plate. Part II, J. Math. Pures Appl., 88 (2007), 149-190.
doi: 10.1016/j.matpur.2007.04.004. |
[12] |
D. Blanchard and G. Griso, Microscopic effects in the homogenization of the junction of rods and a thin plate, Asympt. Anal., 56 (2008), 1-36. |
[13] |
D. Blanchard, A. Gaudiello, T. A. Mel'nyk, Boundary homogenization and reduction of dimension in a Kirchoff-Love plate, SIAM J. Math. Anal., 39 (2008), 1764-1787.
doi: 10.1137/070685919. |
[14] |
R. Brizzi and J. P. Chalot, Boundary homogenization and neumann boundary value problem, Ricerche Mat., 46 (1997), 341-387. |
[15] |
D. Cioranescu and P. Donato, Exact internal controllability in perforated domains, J. Math. Pures. Appl., 68 (1989), 185-213. |
[16] |
D. Cioranescu, P. Donato and E. Zuazua, Exact boundary controllability for the wave equation in domains with small holes, J. Math. Pures. Appl., 71 (1992), 343-377. |
[17] |
A. Corbo Esposito, P. Donato, A. Gaudiello and C. Picard, Homogenization of the p-Laplacian in a domain with oscillating boundary, Comm. Appl. Nonlinear Anal., 4 (1997), 1-23. |
[18] |
A. Damlamian and K. Pettersson, Homogenization of oscillating boundaries, Discrete Contin. Dyn. Syst., 23 (2009), 197-219.
doi: 10.3934/dcds.2009.23.197. |
[19] |
C. D'Apice, U. De Maio and P. I. Kogut, Gap phenomenon in the homogenization of parabolic optimal control problems, IMA J. Math. Control Inform., 25 (2008), 461-489.
doi: 10.1093/imamci/dnn010. |
[20] |
U. De Maio, L. Faella and C. Perugia, Optimal control problem for an anisotropic parabolic problem in a domain with very rough boundary, Ric. Mat., 63 (2014), 307-328.
doi: 10.1007/s11587-014-0183-y. |
[21] |
U. De Maio, L. Faella and C. Perugia, Optimal control for a second-order linear evolution problem in a domain with oscillating boundary, Complex Var. Elliptic Equ., 60 (2015), 1392-1410.
doi: 10.1080/17476933.2015.1022169. |
[22] |
U. De Maio, A. Gaudiello and C. Lefter, Optimal control for a parabolic problem in a domain with higly oscillating boundary, Appl. Anal., 83 (2004), 1245-1264.
doi: 10.1080/00036810410001724670. |
[23] |
U. De Maio and A. K. Nandakumaran, Exact internal controllability for a hyperbolic problem in a domain with highly oscillating boundary, Asympt. Anal., 83 (2013), 189-206. |
[24] |
P. Donato and A. Nabil, Approximate controllability of linear parabolic equations in perforated domain, ESAIM Control Optim. Calc. Var., 6 (2001), 21-38.
doi: 10.1051/cocv:2001102. |
[25] |
T. Durante, L. Faella and C. Perugia, Homogenization and behaviour of optimal controls for the wave equation in domains with oscillating boudary, NoDEA Nonlinear Differential Equations Appl., 14 (2007), 455-489.
doi: 10.1007/s00030-007-3043-6. |
[26] |
T. Durante and T. A. Mel'nyk, Asymptotic analysis of an optimal control problem involving a thick two-level junction with alternate type of controls, J. Optim. Th. and Appl., 144 (2010), 205-225.
doi: 10.1007/s10957-009-9604-6. |
[27] |
L. Faella and C. Perugia, Optimal control for evolutionary imperfect transmission problems, Bound. Value Probl., 2015 (2015), 16pp.
doi: 10.1186/s13661-015-0310-z. |
[28] |
A. Gaudiello, Asymptotic behavior of non-homogeneous Neumann problems in domains with oscillating boundary, Ricerche Mat., 43 (1994), 239-292. |
[29] |
A. Gaudiello, Homogenization of an elliptic trasmission problem, Adv. Math. Sci. Appl., 5 (1995), 639-657. |
[30] |
A. Gaudiello and O. Guibè, Homogenization of an elliptic second-order problem with L log L data in a domain with oscillating boundary, Commun. Contemp. Math., 15 (2013), 1350008, 13pp. |
[31] |
A. Gaudiello, R. Hadiji and C. Picard, Homogenization of the Ginzburg-Landau equation in a domain with oscillating boundary, Commun. Appl. Anal., 7 (2003), 209-223. |
[32] |
J. L. Lions, Controllability Exact, Stabilization at Perturbations de Systéms Distributé, Tomes 1, 2, Masson, Paris, 1988. |
[33] |
J. L. Lions, Exact controllability, stabilization and perturbations for distribuited systems, SIAM Review, 30 (1988), 1-68.
doi: 10.1137/1030001. |
[34] |
J. L. Lions, Contrôlabilité exacte et homogénéisation. I, Asymptotic Analysis, 1 (1988), 3-11. |
[35] |
J. L. Lions and E. Magenes, Problèmes aux limites non homogènes et application, (3 volumes) Dunod, Paris (1968). |
[36] |
J. L. Lions and E. Magenes, Non-homogeneous Boundary Value Problems and Applications, I, II, Springer-Verlag, Berlin Heidelberg, New York, 1972. |
[37] |
T. A. Mel'nyk, Homogenization of the Poisson equation in a thick periodic junction, Z. Anal. Anwendungen, 18 (1999), 953-975.
doi: 10.4171/ZAA/923. |
[38] |
T. A. Mel'nyk, Averaging of a singularly perturbed parabolic problem in a thick periodic junction of the type 3:2:1, Ukrainian Math. J., 52 (2000), 1737-1748.
doi: 10.1023/A:1010483205109. |
[39] |
T. A Mel'nyk and S. A. Nazarov, Asymptotics of the Neumann spectral problem solution in a domain of "thick Comb" type, J. Math. Sci., 85 (1997), 2326-2346.
doi: 10.1007/BF02355841. |
[40] |
F. K. Moore and E. M. Greitzer, A theory of post-stall transients in axial compression systems: Part 1 development of equations, Trans. ASME: J. Eng. Gas Turbines Power, 108 (1986), 68-76.
doi: 10.1115/1.3239887. |
[41] |
F. K. Moore and E. M. Greitzer, A theory of post-stall transients in axial compression systems: Part 2 application, Trans. ASME: J. Eng. Gas Turbines Power, 108 (1986), 231-239. |
[42] |
J. Mossino and A. Sili, Limit behavior of thin heterogeneous domain with rapidly oscillating boundary, Ric. Mat., 56 (2007), 119-148.
doi: 10.1007/s11587-007-0009-2. |
[43] |
A. K. Nandakumaran, Ravi Prakash and J. P. Raymond, Asymptotic analysis and error estimates for an optimal control problem with oscillating boundaries, Annali dell'Università di Ferrara, 58 (2012), 143-166.
doi: 10.1007/s11565-011-0135-3. |
[44] |
A. K. Nandakumaran and R. Prakash, Homogenization of boundary optimal control problems with oscillating boundaries, Nonlinear Studies, 20 (2013), 401-425. |
[45] |
O. Pironneau and C. Saguez, Asymptotic Behaviour, with Respect to the Domains, of Solution of PDE, Laboria Report, 1977. |
[46] |
M. Renardy and R. C. Rogers, An Introduction to Partial Differential Equations, Second edition, Texts in Applied Mathematics, 13, Springer-Verlag, New York, 2004. |
[47] |
J. Simon, Compact sets in the spaces $L^p( 0,T;B)$, Ann. Mat. Pura Appl., 146 (1987), 65-96.
doi: 10.1007/BF01762360. |
[48] |
L. Tartar, Cours Peccot, Collège de France (March 1977), H-Convergence, Séminaire d'analyse fonctionnelle et numérique de l'Université d'Alger (1977-78) (ed. F. MURAT); English translation in Mathematical Modeling of Composite Materials (eds. A. Cherkaev and R. V. Kohon), Progr. Nonlinear Differential Equations Appl., 31, Birkhäuser-Verlag, 1997, 21-43. |
[49] |
E. Zeidler, Nonlinear Functional Analysis and its Applications, Vol. II, Part A and B, Springer-Verlag, Berlin, 1980. |
[50] |
E. Zuazua, Approximate controllability for linear parabolic equations with rapidly oscillating coefficients. Modelling, identification, sensitivity analysis and control of structures, Control and Cybernetics, 23 (1994), 793-801. |
show all references
References:
[1] |
Y. Amirat and O. Bodart, Boundary layer correctors for the solution of laplace equation in a domain with oscillating boundary, Z. Anal. Anwendungen., 20 (2001), 929-940.
doi: 10.4171/ZAA/1052. |
[2] |
Y. Amirat, O. Bodart, U. De Maio and A. Gaudiello, Asymptotic approximation of the solution of the laplace equation in a domain with highly oscillating boundary, SIAM J. Math. Anal., 35 (2004), 1598-1616.
doi: 10.1137/S0036141003414877. |
[3] |
Y. Amirat, O. Bodart, U. De Maio and A. Gaudiello, Asymptotic approximation of the solution of Stokes equation in a domain with highly oscillating boundary, Ann. Univ. Ferrara, 53 (2007), 135-148.
doi: 10.1007/s11565-007-0015-z. |
[4] |
Y. Amirat, O. Bodart, U. De Maio and A. Gaudiello, Effective boundary condition for Stokes flow over a very rough surface, J. Differential Equations, 254 (2013), 3395-3430.
doi: 10.1016/j.jde.2013.01.024. |
[5] |
N. Ansini and A. Braides, Homogenization of oscillating boundaries and applications to thin films, J. Anal. Math., 83 (2001), 151-182.
doi: 10.1007/BF02790260. |
[6] |
V. Barbu and TH. Precupanu, Convexity and Optimization in Banach Spaces, D. Reidel, Dordrecht, 1986. |
[7] |
B. Birnir, S. Hou and N. Wellander, Derivation of the viscous Moore-Greitzer equation for aeroengine flow, J. Math. Phys., 48 (2007), 065209, 31pp.
doi: 10.1063/1.2534332. |
[8] |
D. Blanchard, L. Carbone and A. Gaudiello, Homogenization of a monotone problem in a domain with oscillating boundary, M2AN Math. Model. Numer. Anal., 33 (1999), 1057-1070.
doi: 10.1051/m2an:1999134. |
[9] |
D. Blanchard and A. Gaudiello, Homogenization of highly oscillating boundaries and reduction of dimension for a monotone problem, ESAIM Control Optim. Calc. Var., 9 (2003), 449-460.
doi: 10.1051/cocv:2003022. |
[10] |
D. Blanchard, A. Gaudiello and G. Griso, Junction of a periodic family of elastic rods with a 3d plate. Part I, J. Math. Pures Appl., 88 (2007), 1-33.
doi: 10.1016/j.matpur.2007.04.005. |
[11] |
D. Blanchard, A. Gaudiello and G. Griso, Junction of a periodic family of elastic rods with a thin plate. Part II, J. Math. Pures Appl., 88 (2007), 149-190.
doi: 10.1016/j.matpur.2007.04.004. |
[12] |
D. Blanchard and G. Griso, Microscopic effects in the homogenization of the junction of rods and a thin plate, Asympt. Anal., 56 (2008), 1-36. |
[13] |
D. Blanchard, A. Gaudiello, T. A. Mel'nyk, Boundary homogenization and reduction of dimension in a Kirchoff-Love plate, SIAM J. Math. Anal., 39 (2008), 1764-1787.
doi: 10.1137/070685919. |
[14] |
R. Brizzi and J. P. Chalot, Boundary homogenization and neumann boundary value problem, Ricerche Mat., 46 (1997), 341-387. |
[15] |
D. Cioranescu and P. Donato, Exact internal controllability in perforated domains, J. Math. Pures. Appl., 68 (1989), 185-213. |
[16] |
D. Cioranescu, P. Donato and E. Zuazua, Exact boundary controllability for the wave equation in domains with small holes, J. Math. Pures. Appl., 71 (1992), 343-377. |
[17] |
A. Corbo Esposito, P. Donato, A. Gaudiello and C. Picard, Homogenization of the p-Laplacian in a domain with oscillating boundary, Comm. Appl. Nonlinear Anal., 4 (1997), 1-23. |
[18] |
A. Damlamian and K. Pettersson, Homogenization of oscillating boundaries, Discrete Contin. Dyn. Syst., 23 (2009), 197-219.
doi: 10.3934/dcds.2009.23.197. |
[19] |
C. D'Apice, U. De Maio and P. I. Kogut, Gap phenomenon in the homogenization of parabolic optimal control problems, IMA J. Math. Control Inform., 25 (2008), 461-489.
doi: 10.1093/imamci/dnn010. |
[20] |
U. De Maio, L. Faella and C. Perugia, Optimal control problem for an anisotropic parabolic problem in a domain with very rough boundary, Ric. Mat., 63 (2014), 307-328.
doi: 10.1007/s11587-014-0183-y. |
[21] |
U. De Maio, L. Faella and C. Perugia, Optimal control for a second-order linear evolution problem in a domain with oscillating boundary, Complex Var. Elliptic Equ., 60 (2015), 1392-1410.
doi: 10.1080/17476933.2015.1022169. |
[22] |
U. De Maio, A. Gaudiello and C. Lefter, Optimal control for a parabolic problem in a domain with higly oscillating boundary, Appl. Anal., 83 (2004), 1245-1264.
doi: 10.1080/00036810410001724670. |
[23] |
U. De Maio and A. K. Nandakumaran, Exact internal controllability for a hyperbolic problem in a domain with highly oscillating boundary, Asympt. Anal., 83 (2013), 189-206. |
[24] |
P. Donato and A. Nabil, Approximate controllability of linear parabolic equations in perforated domain, ESAIM Control Optim. Calc. Var., 6 (2001), 21-38.
doi: 10.1051/cocv:2001102. |
[25] |
T. Durante, L. Faella and C. Perugia, Homogenization and behaviour of optimal controls for the wave equation in domains with oscillating boudary, NoDEA Nonlinear Differential Equations Appl., 14 (2007), 455-489.
doi: 10.1007/s00030-007-3043-6. |
[26] |
T. Durante and T. A. Mel'nyk, Asymptotic analysis of an optimal control problem involving a thick two-level junction with alternate type of controls, J. Optim. Th. and Appl., 144 (2010), 205-225.
doi: 10.1007/s10957-009-9604-6. |
[27] |
L. Faella and C. Perugia, Optimal control for evolutionary imperfect transmission problems, Bound. Value Probl., 2015 (2015), 16pp.
doi: 10.1186/s13661-015-0310-z. |
[28] |
A. Gaudiello, Asymptotic behavior of non-homogeneous Neumann problems in domains with oscillating boundary, Ricerche Mat., 43 (1994), 239-292. |
[29] |
A. Gaudiello, Homogenization of an elliptic trasmission problem, Adv. Math. Sci. Appl., 5 (1995), 639-657. |
[30] |
A. Gaudiello and O. Guibè, Homogenization of an elliptic second-order problem with L log L data in a domain with oscillating boundary, Commun. Contemp. Math., 15 (2013), 1350008, 13pp. |
[31] |
A. Gaudiello, R. Hadiji and C. Picard, Homogenization of the Ginzburg-Landau equation in a domain with oscillating boundary, Commun. Appl. Anal., 7 (2003), 209-223. |
[32] |
J. L. Lions, Controllability Exact, Stabilization at Perturbations de Systéms Distributé, Tomes 1, 2, Masson, Paris, 1988. |
[33] |
J. L. Lions, Exact controllability, stabilization and perturbations for distribuited systems, SIAM Review, 30 (1988), 1-68.
doi: 10.1137/1030001. |
[34] |
J. L. Lions, Contrôlabilité exacte et homogénéisation. I, Asymptotic Analysis, 1 (1988), 3-11. |
[35] |
J. L. Lions and E. Magenes, Problèmes aux limites non homogènes et application, (3 volumes) Dunod, Paris (1968). |
[36] |
J. L. Lions and E. Magenes, Non-homogeneous Boundary Value Problems and Applications, I, II, Springer-Verlag, Berlin Heidelberg, New York, 1972. |
[37] |
T. A. Mel'nyk, Homogenization of the Poisson equation in a thick periodic junction, Z. Anal. Anwendungen, 18 (1999), 953-975.
doi: 10.4171/ZAA/923. |
[38] |
T. A. Mel'nyk, Averaging of a singularly perturbed parabolic problem in a thick periodic junction of the type 3:2:1, Ukrainian Math. J., 52 (2000), 1737-1748.
doi: 10.1023/A:1010483205109. |
[39] |
T. A Mel'nyk and S. A. Nazarov, Asymptotics of the Neumann spectral problem solution in a domain of "thick Comb" type, J. Math. Sci., 85 (1997), 2326-2346.
doi: 10.1007/BF02355841. |
[40] |
F. K. Moore and E. M. Greitzer, A theory of post-stall transients in axial compression systems: Part 1 development of equations, Trans. ASME: J. Eng. Gas Turbines Power, 108 (1986), 68-76.
doi: 10.1115/1.3239887. |
[41] |
F. K. Moore and E. M. Greitzer, A theory of post-stall transients in axial compression systems: Part 2 application, Trans. ASME: J. Eng. Gas Turbines Power, 108 (1986), 231-239. |
[42] |
J. Mossino and A. Sili, Limit behavior of thin heterogeneous domain with rapidly oscillating boundary, Ric. Mat., 56 (2007), 119-148.
doi: 10.1007/s11587-007-0009-2. |
[43] |
A. K. Nandakumaran, Ravi Prakash and J. P. Raymond, Asymptotic analysis and error estimates for an optimal control problem with oscillating boundaries, Annali dell'Università di Ferrara, 58 (2012), 143-166.
doi: 10.1007/s11565-011-0135-3. |
[44] |
A. K. Nandakumaran and R. Prakash, Homogenization of boundary optimal control problems with oscillating boundaries, Nonlinear Studies, 20 (2013), 401-425. |
[45] |
O. Pironneau and C. Saguez, Asymptotic Behaviour, with Respect to the Domains, of Solution of PDE, Laboria Report, 1977. |
[46] |
M. Renardy and R. C. Rogers, An Introduction to Partial Differential Equations, Second edition, Texts in Applied Mathematics, 13, Springer-Verlag, New York, 2004. |
[47] |
J. Simon, Compact sets in the spaces $L^p( 0,T;B)$, Ann. Mat. Pura Appl., 146 (1987), 65-96.
doi: 10.1007/BF01762360. |
[48] |
L. Tartar, Cours Peccot, Collège de France (March 1977), H-Convergence, Séminaire d'analyse fonctionnelle et numérique de l'Université d'Alger (1977-78) (ed. F. MURAT); English translation in Mathematical Modeling of Composite Materials (eds. A. Cherkaev and R. V. Kohon), Progr. Nonlinear Differential Equations Appl., 31, Birkhäuser-Verlag, 1997, 21-43. |
[49] |
E. Zeidler, Nonlinear Functional Analysis and its Applications, Vol. II, Part A and B, Springer-Verlag, Berlin, 1980. |
[50] |
E. Zuazua, Approximate controllability for linear parabolic equations with rapidly oscillating coefficients. Modelling, identification, sensitivity analysis and control of structures, Control and Cybernetics, 23 (1994), 793-801. |
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