# American Institute of Mathematical Sciences

September  2015, 4(3): 325-346. doi: 10.3934/eect.2015.4.325

## 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

Received  December 2014 Revised  April 2015 Published  September 2015

In this paper, we study the exact controllability of a second order linear evolution equation in a domain with highly oscillating boundary with homogeneous Neumann boundary condition on the oscillating part of boundary. Our aim is to obtain the exact controllability for the homogenized equation. The limit problem with Neumann condition on the oscillating boundary is different and hence we need to study the exact controllability of this new type of problem. In the process of homogenization, we also study the asymptotic analysis of evolution equation in two setups, namely solution by standard weak formulation and solution by transposition method.
Citation: Umberto De Maio, Akamabadath K. Nandakumaran, Carmen Perugia. Exact internal controllability for the wave equation in a domain with oscillating boundary with Neumann boundary condition. Evolution Equations and Control Theory, 2015, 4 (3) : 325-346. doi: 10.3934/eect.2015.4.325
##### 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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