# 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 & 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. doi: 10.4171/ZAA/1052. Google Scholar [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. doi: 10.1137/S0036141003414877. Google Scholar [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. doi: 10.1007/s11565-007-0015-z. Google Scholar [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. doi: 10.1016/j.jde.2013.01.024. Google Scholar [5] N. Ansini and A. Braides, Homogenization of oscillating boundaries and applications to thin films,, J. Anal. Math., 83 (2001), 151. doi: 10.1007/BF02790260. Google Scholar [6] V. Barbu and TH. Precupanu, Convexity and Optimization in Banach Spaces,, D. Reidel, (1986). Google Scholar [7] B. Birnir, S. Hou and N. Wellander, Derivation of the viscous Moore-Greitzer equation for aeroengine flow,, J. Math. Phys., 48 (2007). doi: 10.1063/1.2534332. Google Scholar [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. doi: 10.1051/m2an:1999134. Google Scholar [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. doi: 10.1051/cocv:2003022. Google Scholar [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. doi: 10.1016/j.matpur.2007.04.005. Google Scholar [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. doi: 10.1016/j.matpur.2007.04.004. Google Scholar [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. Google Scholar [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. doi: 10.1137/070685919. Google Scholar [14] R. Brizzi and J. P. Chalot, Boundary homogenization and neumann boundary value problem,, Ricerche Mat., 46 (1997), 341. Google Scholar [15] D. Cioranescu and P. Donato, Exact internal controllability in perforated domains,, J. Math. Pures. Appl., 68 (1989), 185. Google Scholar [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. Google Scholar [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. Google Scholar [18] A. Damlamian and K. Pettersson, Homogenization of oscillating boundaries,, Discrete Contin. Dyn. Syst., 23 (2009), 197. doi: 10.3934/dcds.2009.23.197. Google Scholar [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. doi: 10.1093/imamci/dnn010. Google Scholar [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. doi: 10.1007/s11587-014-0183-y. Google Scholar [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. doi: 10.1080/17476933.2015.1022169. Google Scholar [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. doi: 10.1080/00036810410001724670. Google Scholar [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. Google Scholar [24] P. Donato and A. Nabil, Approximate controllability of linear parabolic equations in perforated domain,, ESAIM Control Optim. Calc. Var., 6 (2001), 21. doi: 10.1051/cocv:2001102. Google Scholar [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. doi: 10.1007/s00030-007-3043-6. Google Scholar [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. doi: 10.1007/s10957-009-9604-6. Google Scholar [27] L. Faella and C. Perugia, Optimal control for evolutionary imperfect transmission problems,, Bound. Value Probl., 2015 (2015). doi: 10.1186/s13661-015-0310-z. Google Scholar [28] A. Gaudiello, Asymptotic behavior of non-homogeneous Neumann problems in domains with oscillating boundary,, Ricerche Mat., 43 (1994), 239. Google Scholar [29] A. Gaudiello, Homogenization of an elliptic trasmission problem,, Adv. Math. Sci. Appl., 5 (1995), 639. Google Scholar [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). Google Scholar [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. Google Scholar [32] J. L. Lions, Controllability Exact, Stabilization at Perturbations de Systéms Distributé, Tomes 1, 2,, Masson, (1988). Google Scholar [33] J. L. Lions, Exact controllability, stabilization and perturbations for distribuited systems,, SIAM Review, 30 (1988), 1. doi: 10.1137/1030001. Google Scholar [34] J. L. Lions, Contrôlabilité exacte et homogénéisation. I,, Asymptotic Analysis, 1 (1988), 3. Google Scholar [35] J. L. Lions and E. Magenes, Problèmes aux limites non homogènes et application, (3 volumes), Dunod, (1968). Google Scholar [36] J. L. Lions and E. Magenes, Non-homogeneous Boundary Value Problems and Applications, I, II,, Springer-Verlag, (1972). Google Scholar [37] T. A. Mel'nyk, Homogenization of the Poisson equation in a thick periodic junction,, Z. Anal. Anwendungen, 18 (1999), 953. doi: 10.4171/ZAA/923. Google Scholar [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. doi: 10.1023/A:1010483205109. Google Scholar [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. doi: 10.1007/BF02355841. Google Scholar [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. doi: 10.1115/1.3239887. Google Scholar [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. Google Scholar [42] J. Mossino and A. Sili, Limit behavior of thin heterogeneous domain with rapidly oscillating boundary,, Ric. Mat., 56 (2007), 119. doi: 10.1007/s11587-007-0009-2. Google Scholar [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. doi: 10.1007/s11565-011-0135-3. Google Scholar [44] A. K. Nandakumaran and R. Prakash, Homogenization of boundary optimal control problems with oscillating boundaries,, Nonlinear Studies, 20 (2013), 401. Google Scholar [45] O. Pironneau and C. Saguez, Asymptotic Behaviour, with Respect to the Domains, of Solution of PDE,, Laboria Report, (1977). Google Scholar [46] M. Renardy and R. C. Rogers, An Introduction to Partial Differential Equations,, Second edition, (2004). Google Scholar [47] J. Simon, Compact sets in the spaces $L^p( 0,T;B)$,, Ann. Mat. Pura Appl., 146 (1987), 65. doi: 10.1007/BF01762360. Google Scholar [48] L. Tartar, Cours Peccot, Collège de France (March 1977),, H-Convergence, (1997), 1977. Google Scholar [49] E. Zeidler, Nonlinear Functional Analysis and its Applications, Vol. II, Part A and B,, Springer-Verlag, (1980). Google Scholar [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. Google Scholar

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. doi: 10.4171/ZAA/1052. Google Scholar [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. doi: 10.1137/S0036141003414877. Google Scholar [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. doi: 10.1007/s11565-007-0015-z. Google Scholar [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. doi: 10.1016/j.jde.2013.01.024. Google Scholar [5] N. Ansini and A. Braides, Homogenization of oscillating boundaries and applications to thin films,, J. Anal. Math., 83 (2001), 151. doi: 10.1007/BF02790260. Google Scholar [6] V. Barbu and TH. Precupanu, Convexity and Optimization in Banach Spaces,, D. Reidel, (1986). Google Scholar [7] B. Birnir, S. Hou and N. Wellander, Derivation of the viscous Moore-Greitzer equation for aeroengine flow,, J. Math. Phys., 48 (2007). doi: 10.1063/1.2534332. Google Scholar [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. doi: 10.1051/m2an:1999134. Google Scholar [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. doi: 10.1051/cocv:2003022. Google Scholar [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. doi: 10.1016/j.matpur.2007.04.005. Google Scholar [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. doi: 10.1016/j.matpur.2007.04.004. Google Scholar [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. Google Scholar [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. doi: 10.1137/070685919. Google Scholar [14] R. Brizzi and J. P. Chalot, Boundary homogenization and neumann boundary value problem,, Ricerche Mat., 46 (1997), 341. Google Scholar [15] D. Cioranescu and P. Donato, Exact internal controllability in perforated domains,, J. Math. Pures. Appl., 68 (1989), 185. Google Scholar [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. Google Scholar [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. Google Scholar [18] A. Damlamian and K. Pettersson, Homogenization of oscillating boundaries,, Discrete Contin. Dyn. Syst., 23 (2009), 197. doi: 10.3934/dcds.2009.23.197. Google Scholar [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. doi: 10.1093/imamci/dnn010. Google Scholar [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. doi: 10.1007/s11587-014-0183-y. Google Scholar [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. doi: 10.1080/17476933.2015.1022169. Google Scholar [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. doi: 10.1080/00036810410001724670. Google Scholar [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. Google Scholar [24] P. Donato and A. Nabil, Approximate controllability of linear parabolic equations in perforated domain,, ESAIM Control Optim. Calc. Var., 6 (2001), 21. doi: 10.1051/cocv:2001102. Google Scholar [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. doi: 10.1007/s00030-007-3043-6. Google Scholar [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. doi: 10.1007/s10957-009-9604-6. Google Scholar [27] L. Faella and C. Perugia, Optimal control for evolutionary imperfect transmission problems,, Bound. Value Probl., 2015 (2015). doi: 10.1186/s13661-015-0310-z. Google Scholar [28] A. Gaudiello, Asymptotic behavior of non-homogeneous Neumann problems in domains with oscillating boundary,, Ricerche Mat., 43 (1994), 239. Google Scholar [29] A. Gaudiello, Homogenization of an elliptic trasmission problem,, Adv. Math. Sci. Appl., 5 (1995), 639. Google Scholar [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). Google Scholar [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. Google Scholar [32] J. L. Lions, Controllability Exact, Stabilization at Perturbations de Systéms Distributé, Tomes 1, 2,, Masson, (1988). Google Scholar [33] J. L. Lions, Exact controllability, stabilization and perturbations for distribuited systems,, SIAM Review, 30 (1988), 1. doi: 10.1137/1030001. Google Scholar [34] J. L. Lions, Contrôlabilité exacte et homogénéisation. I,, Asymptotic Analysis, 1 (1988), 3. Google Scholar [35] J. L. Lions and E. Magenes, Problèmes aux limites non homogènes et application, (3 volumes), Dunod, (1968). Google Scholar [36] J. L. Lions and E. Magenes, Non-homogeneous Boundary Value Problems and Applications, I, II,, Springer-Verlag, (1972). Google Scholar [37] T. A. Mel'nyk, Homogenization of the Poisson equation in a thick periodic junction,, Z. Anal. Anwendungen, 18 (1999), 953. doi: 10.4171/ZAA/923. Google Scholar [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. doi: 10.1023/A:1010483205109. Google Scholar [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. doi: 10.1007/BF02355841. Google Scholar [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. doi: 10.1115/1.3239887. Google Scholar [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. Google Scholar [42] J. Mossino and A. Sili, Limit behavior of thin heterogeneous domain with rapidly oscillating boundary,, Ric. Mat., 56 (2007), 119. doi: 10.1007/s11587-007-0009-2. Google Scholar [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. doi: 10.1007/s11565-011-0135-3. Google Scholar [44] A. K. Nandakumaran and R. Prakash, Homogenization of boundary optimal control problems with oscillating boundaries,, Nonlinear Studies, 20 (2013), 401. Google Scholar [45] O. Pironneau and C. Saguez, Asymptotic Behaviour, with Respect to the Domains, of Solution of PDE,, Laboria Report, (1977). Google Scholar [46] M. Renardy and R. C. Rogers, An Introduction to Partial Differential Equations,, Second edition, (2004). Google Scholar [47] J. Simon, Compact sets in the spaces $L^p( 0,T;B)$,, Ann. Mat. Pura Appl., 146 (1987), 65. doi: 10.1007/BF01762360. Google Scholar [48] L. Tartar, Cours Peccot, Collège de France (March 1977),, H-Convergence, (1997), 1977. Google Scholar [49] E. Zeidler, Nonlinear Functional Analysis and its Applications, Vol. II, Part A and B,, Springer-Verlag, (1980). Google Scholar [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. Google Scholar
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