June  2019, 14(2): 411-444. doi: 10.3934/nhm.2019017

Homogenization and exact controllability for problems with imperfect interface

1. 

Dipartimento di Matematica, Università di Salerno, via Giovanni Paolo Ⅱ, 132, 84084, Fisciano (SA), Italy

2. 

Dipartimento di Scienze e Tecnologie, Università del Sannio, Via Port'Arsa, 11, 82100, Benevento (BN), Italy

* Corresponding author: Sara Monsurrò

Received  November 2018 Published  April 2019

The first aim of this paper is to study, by means of the periodic unfolding method, the homogenization of elliptic problems with source terms converging in a space of functions less regular than the usual $ L^2 $, in an $ \varepsilon $-periodic two component composite with an imperfect transmission condition on the interface. Then we exploit this result to describe the asymptotic behaviour of the exact controls and the corresponding states of hyperbolic problems set in composites with the same structure and presenting the same condition on the interface. The exact controllability is developed by applying the Hilbert Uniqueness Method, introduced by J. -L. Lions, which leads us to the construction of the exact controls as solutions of suitable transposed problem.

Citation: Sara Monsurrò, Carmen Perugia. Homogenization and exact controllability for problems with imperfect interface. Networks & Heterogeneous Media, 2019, 14 (2) : 411-444. doi: 10.3934/nhm.2019017
References:
[1]

J. L. Auriault and H. Ene, Macroscopic modelling of heat transfer in composites with interfacial thermal barrier, Internat. J. Heat Mass Transfer, 37 (1994), 2885-2892. doi: 10.1016/0017-9310(94)90342-5. Google Scholar

[2]

A. Bensoussan, J. L. Lions and G. Papanicolaou, Asymptotic Analysis for Periodic Structures, North-Holland, Amsterdam, 1978. Google Scholar

[3]

E. Canon and J. N. Pernin, Homogenization of diffusion in composite media with interfacial barrier, Rev. Roumaine Math. Pures Appl., 44 (1999), 23-36. Google Scholar

[4]

G. CardoneS. E. Pastukhova and C. Perugia, Estimates in homogenization of degenerate elliptic equations by spectral method, Asymptot. Anal., 81 (2013), 189-209. Google Scholar

[5] H. S. Carslaw and J. C. Jaeger, Conduction of Heat in Solids, At the Clarendon Press, Oxford, 1947. Google Scholar
[6]

D. Cioranescu, A. Damlamian and G. Griso, The periodic unfolding method in homogenization, SIAM J. Math. Anal., 40 (2008), 1585–1620. doi: 10.1137/080713148. Google Scholar

[7]

D. CioranescuA. DamlamianP. DonatoG. Griso and R. Zaki, The periodic unfolding method in domains with holes, SIAM J. Math. Anal., 44 (2012), 718-760. doi: 10.1137/100817942. Google Scholar

[8]

D. Cioranescu and P. Donato, Exact internal controllability in perforated domains, J. Math. Pures. Appl, 68 (1989), 185-213. Google Scholar

[9] D. Cioranescu and P. Donato, An Introduction to Homogenization, Oxford Lecture Ser. Math., Appl. 17, Oxford University Press, New York, 1999. Google Scholar
[10]

D. CioranescuP. Donato and R. Zaki, The periodic unfolding method in perforated domains, Portugaliae Mathematica, 63 (2006), 467-496. Google Scholar

[11]

D. CioranescuP. Donato and E. Zuazua, Exact boundary controllability for the wave equation in domains with small holes, J. Math. Pures. Appl., 71 (1992), 343-377. Google Scholar

[12]

D. Cioranescu and J. Saint Jean Paulin, Homogenization in open sets with holes, J. Math. Anal. Appl., 71 (1979), 590-607. doi: 10.1016/0022-247X(79)90211-7. Google Scholar

[13]

G. Dal Maso and F. Murat, Asymptotic behaviour and correctors for linear Dirichlet problems with simultaneously varying operators and domains, Ann. I. H. Poincaré, 21 (2004), 445-486. doi: 10.1016/j.anihpc.2003.05.001. Google Scholar

[14]

U. De MaioA. 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. Google Scholar

[15]

U. De MaioL. 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. Google Scholar

[16]

U. De MaioL. Faella and C. Perugia, Optimal control for a second-order linear evolution problem in a domain with oscillating boundary, Complex Var. Elliptic Equ., 6 (2015), 1392-1410. doi: 10.1080/17476933.2015.1022169. Google Scholar

[17]

U. De Maio and A. K. Nandakumaran, Exact internal controllability for a hyperbolic problem in a domain with highly oscillating boundary, Asymptot. Anal., 83 (2013), 189-206. Google Scholar

[18]

U. De MaioA. K. Nandakumaran and C. Perugia, Exact internal controllability for the wave equation in a domain with oscillating boundary with Neumann boundary condition, Evol. Equ. Control Theory, 4 (2015), 325-346. doi: 10.3934/eect.2015.4.325. Google Scholar

[19]

P. Donato, Some corrector results for composites with imperfect interface, Rend. Mat. Ser. Ⅶ, 26 (2006), 189-209. Google Scholar

[20]

P. Donato, Homogenization of a class of imperfect transmission problems, in Multiscale Problems: Theory, Numerical Approximation and Applications doi: 10.1142/9789814366892_0001. Google Scholar

[21]

P. DonatoL. Faella and S. Monsurrò, Homogenization of the wave equation in composites with imperfect interface: A memory effect, J. Math. Pures Appl., 87 (2007), 119-143. doi: 10.1016/j.matpur.2006.11.004. Google Scholar

[22]

P. DonatoL. Faella and S. Monsurrò, Correctors for the homogenization of a class of hyperbolic equations with imperfect interfaces, SIAM J. Math. Anal., 40 (2009), 1952-1978. doi: 10.1137/080712684. Google Scholar

[23]

P. Donato and E. Jose, Corrector results for a parabolic problem with a memory effect, ESAIM: Math. Model. Numer. Anal., 44 (2010), 421-454. doi: 10.1051/m2an/2010008. Google Scholar

[24]

P. Donato and E. Jose, Asymptotic behavior of the approximate controls for parabolic equations with interfacial contact resistance, ESAIM Control Optim. Calc. Var., 21 (2015), 138-164. doi: 10.1051/cocv/2014029. Google Scholar

[25]

P. Donato and E. Jose, Approximate controllability of a parabolic system with imperfect interfaces, Philipp. J. Sci., 144 (2015), 187-196. Google Scholar

[26]

P. DonatoK. H. Le Nguyen and R. Tardieu, The periodic unfolding method for a class of imperfect trasmission problems, J. Math. Sci., 176 (2011), 891-927. doi: 10.1007/s10958-011-0443-2. Google Scholar

[27]

P. Donato and K. H. Le Nguyen, Homogenization of diffusion problems with a nonlinear interfacial resistance, Nonlinear Differ. Equ. Appl., 22 (2015), 1345-1380. doi: 10.1007/s00030-015-0325-2. Google Scholar

[28]

P. Donato and S. Monsurrò, Homogenization of two heat conductors with interfacial contact resistance, Anal. Appl., 2 (2004), 247-273. doi: 10.1142/S0219530504000345. Google Scholar

[29]

P. DonatoS. Monsurrò and F. Raimondi, Homogenization of a class of singular elliptic problems in perforated domains, Nonlinear Anal., 173 (2018), 180-208. doi: 10.1016/j.na.2018.04.005. Google Scholar

[30]

P. Donato and A. Nabil, Approximate controllability of linear parabolic equations in perforated domains, ESAIM Control Optim. Calc. Var., 6 (2001), 21-38. doi: 10.1051/cocv:2001102. Google Scholar

[31]

T. DuranteL. Faella and C. Perugia, Homogenization and behaviour of optimal controls for the wave equation in domains with oscillating boudary, Nonlinear Differ. Equ. Appl., 14 (2007), 455-489. doi: 10.1007/s00030-007-3043-6. Google Scholar

[32]

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. Google Scholar

[33]

T. Durante and T. A. Mel'nyk, Homogenization of quasilinear optimal control problems involving a thick multilevel junction of type 3:2:1, ESAIM Control Optim. Calc. Var., 18 (2012), 583-610. doi: 10.1051/cocv/2011107. Google Scholar

[34]

L. Faella and S. Monsurrò, Memory effects arising in the homogenization of composites with inclusions, in Topics on Mathematics for Smart System doi: 10.1142/9789812706874_0008. Google Scholar

[35]

L. FaellaS. Monsurrò and C. Perugia, Homogenization of imperfect transmission problems: The case of weakly converging data, Differential Integral Equations, 31 (2018), 595-620. Google Scholar

[36]

L. FaellaS. Monsurrò and C. Perugia, Exact controllability for an imperfect transmission problem, J. Math. Pures Appl., 122 (2019), 235-271. doi: 10.1016/j.matpur.2017.11.011. Google Scholar

[37]

L. Faella and C. Perugia, Homogenization of a Ginzburg-Landau problem in a perforated domain with mixed boundary conditions, Bound. Value Probl., 2014 (2014), 28pp. doi: 10.1186/s13661-014-0223-2. Google Scholar

[38]

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. Google Scholar

[39]

L. Faella and C. Perugia, Optimal control for a hyperbolic problem in composites with imperfect interface: A memory effect, Evol. Equ. Control Theory, 6 (2017), 187-217. doi: 10.3934/eect.2017011. Google Scholar

[40]

E. Fernandez-Cara, Null controllability of the semilinear heat equation, ESAIM Control Optim. Calc. Var., 2 (1997), 87-103. doi: 10.1051/cocv:1997104. Google Scholar

[41]

H. C. Hummel, Homogenization for heat transfer in polycristals with interfacial resistances, Appl. Anal., 75 (2000), 403-424. doi: 10.1080/00036810008840857. Google Scholar

[42]

A. M. Khludnev, L. Faella and C. Perugia, Optimal control of rigidity parameters of thin inclusions in composite materials, Z. Angew. Math. Phys., 68 (2017), Art. 47, 12 pp. doi: 10.1007/s00033-017-0792-x. Google Scholar

[43]

L. Li and X. Zhang, Exact controllability for semilinear wave equations, J. Math. Anal. Appl., 250 (2000), 589-597. doi: 10.1006/jmaa.2000.6998. Google Scholar

[44]

J. L. Lions, Contrôlabilité Exacte et Homogénéisation, I, Asymptotic Anal., 1 (1988), 3-11. Google Scholar

[45]

J. L. Lions, Contrôlabilité Exacte, Stabilization at Perturbations De Systéms Distributé, Tomes 1, 2, Massonn, RMA, 829, 1988. Google Scholar

[46]

J. L. Lions and E. Magenes, Non-homogeneous Boundary Value Problems and Applications, Vol I, Springer-Verlag Berlin Heidelberg, New York, 1972. Google Scholar

[47]

R. Lipton, Heat conduction in fine scale mixtures with interfacial contact resistance, SIAM J. Appl. Math., 58 (1998), 55-72. doi: 10.1137/S0036139995295153. Google Scholar

[48]

R. Lipton and B. Vernescu, Composite with imperfect interface, Proc. R. Soc. Lond. Ser. A, 452 (1996), 329-358. doi: 10.1098/rspa.1996.0018. Google Scholar

[49]

S. Monsurrò, Homogenization of a two-component composite with interfacial thermal barrier, Adv. Math. Sci. Appl., 13 (2003), 43-63. Google Scholar

[50]

S. Monsurrò, Erratum for the paper "Homogenization of a two-component composite with interfacial thermal barrier", Adv. Math. Sci. Appl., 14 (2004), 375-377. Google Scholar

[51]

S. Monsurrò, Homogenization of a composite with very small inclusions and imperfect interface, in Multiscale problems and asymptotic analysis, GAKUTO Internat. Ser. Math. Sci. Appl., 24, Gakkotosho, Tokyo, (2006), 217–232. Google Scholar

[52]

T. Muthukumar and A. K. Nandakumaran, Homogenization of low-cost control problems on perforated domains, J. Math. Anal. Appl., 351 (2009), 29-42. doi: 10.1016/j.jmaa.2008.09.048. Google Scholar

[53]

A.K. NandakumaranR. Prakash and B. C. Sardar, Asymptotic analysis of Neumann periodic optimal boundary control problem, Math. Methods Appl. Sci., 39 (2016), 4354-4374. doi: 10.1002/mma.3865. Google Scholar

[54]

Z. Yang, Homogenization and correctors for the hyperbolic problems with imperfect interfaces via the periodic unfolding method, Commun. Pure Appl. Anal., 13 (2014), 249-272. doi: 10.3934/cpaa.2014.13.249. Google Scholar

[55]

Z. Yang, The periodic unfolding method for a class of parabolic problems with imperfect interfaces, ESAIM Math. Model. Numer. Anal., 48 (2014), 1279-1302. doi: 10.1051/m2an/2013139. Google Scholar

[56]

E. Zuazua, Exact boundary controllability for the semilinear wave equation, in Nonlinear Partial Differential Equations and Their Applications, Collège de France Seminar, Vol. X (Paris, 1987–1988), Pitman Res. Notes Math. Ser., 220, Longman Sci. Tech., Harlow, 1991, 357–391. Google Scholar

[57]

E. Zuazua, Approximate controllability for linear parabolic equations with rapidly oscillating coefficients, Control Cybernet., 23 (1994), 793-801. Google Scholar

[58]

E. Zuazua, Controllability of partial differential equations and its semi-discrete approximations, Discrete Contin. Dyn. Syst., 8 (2002), 469-513. doi: 10.3934/dcds.2002.8.469. Google Scholar

show all references

References:
[1]

J. L. Auriault and H. Ene, Macroscopic modelling of heat transfer in composites with interfacial thermal barrier, Internat. J. Heat Mass Transfer, 37 (1994), 2885-2892. doi: 10.1016/0017-9310(94)90342-5. Google Scholar

[2]

A. Bensoussan, J. L. Lions and G. Papanicolaou, Asymptotic Analysis for Periodic Structures, North-Holland, Amsterdam, 1978. Google Scholar

[3]

E. Canon and J. N. Pernin, Homogenization of diffusion in composite media with interfacial barrier, Rev. Roumaine Math. Pures Appl., 44 (1999), 23-36. Google Scholar

[4]

G. CardoneS. E. Pastukhova and C. Perugia, Estimates in homogenization of degenerate elliptic equations by spectral method, Asymptot. Anal., 81 (2013), 189-209. Google Scholar

[5] H. S. Carslaw and J. C. Jaeger, Conduction of Heat in Solids, At the Clarendon Press, Oxford, 1947. Google Scholar
[6]

D. Cioranescu, A. Damlamian and G. Griso, The periodic unfolding method in homogenization, SIAM J. Math. Anal., 40 (2008), 1585–1620. doi: 10.1137/080713148. Google Scholar

[7]

D. CioranescuA. DamlamianP. DonatoG. Griso and R. Zaki, The periodic unfolding method in domains with holes, SIAM J. Math. Anal., 44 (2012), 718-760. doi: 10.1137/100817942. Google Scholar

[8]

D. Cioranescu and P. Donato, Exact internal controllability in perforated domains, J. Math. Pures. Appl, 68 (1989), 185-213. Google Scholar

[9] D. Cioranescu and P. Donato, An Introduction to Homogenization, Oxford Lecture Ser. Math., Appl. 17, Oxford University Press, New York, 1999. Google Scholar
[10]

D. CioranescuP. Donato and R. Zaki, The periodic unfolding method in perforated domains, Portugaliae Mathematica, 63 (2006), 467-496. Google Scholar

[11]

D. CioranescuP. Donato and E. Zuazua, Exact boundary controllability for the wave equation in domains with small holes, J. Math. Pures. Appl., 71 (1992), 343-377. Google Scholar

[12]

D. Cioranescu and J. Saint Jean Paulin, Homogenization in open sets with holes, J. Math. Anal. Appl., 71 (1979), 590-607. doi: 10.1016/0022-247X(79)90211-7. Google Scholar

[13]

G. Dal Maso and F. Murat, Asymptotic behaviour and correctors for linear Dirichlet problems with simultaneously varying operators and domains, Ann. I. H. Poincaré, 21 (2004), 445-486. doi: 10.1016/j.anihpc.2003.05.001. Google Scholar

[14]

U. De MaioA. 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. Google Scholar

[15]

U. De MaioL. 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. Google Scholar

[16]

U. De MaioL. Faella and C. Perugia, Optimal control for a second-order linear evolution problem in a domain with oscillating boundary, Complex Var. Elliptic Equ., 6 (2015), 1392-1410. doi: 10.1080/17476933.2015.1022169. Google Scholar

[17]

U. De Maio and A. K. Nandakumaran, Exact internal controllability for a hyperbolic problem in a domain with highly oscillating boundary, Asymptot. Anal., 83 (2013), 189-206. Google Scholar

[18]

U. De MaioA. K. Nandakumaran and C. Perugia, Exact internal controllability for the wave equation in a domain with oscillating boundary with Neumann boundary condition, Evol. Equ. Control Theory, 4 (2015), 325-346. doi: 10.3934/eect.2015.4.325. Google Scholar

[19]

P. Donato, Some corrector results for composites with imperfect interface, Rend. Mat. Ser. Ⅶ, 26 (2006), 189-209. Google Scholar

[20]

P. Donato, Homogenization of a class of imperfect transmission problems, in Multiscale Problems: Theory, Numerical Approximation and Applications doi: 10.1142/9789814366892_0001. Google Scholar

[21]

P. DonatoL. Faella and S. Monsurrò, Homogenization of the wave equation in composites with imperfect interface: A memory effect, J. Math. Pures Appl., 87 (2007), 119-143. doi: 10.1016/j.matpur.2006.11.004. Google Scholar

[22]

P. DonatoL. Faella and S. Monsurrò, Correctors for the homogenization of a class of hyperbolic equations with imperfect interfaces, SIAM J. Math. Anal., 40 (2009), 1952-1978. doi: 10.1137/080712684. Google Scholar

[23]

P. Donato and E. Jose, Corrector results for a parabolic problem with a memory effect, ESAIM: Math. Model. Numer. Anal., 44 (2010), 421-454. doi: 10.1051/m2an/2010008. Google Scholar

[24]

P. Donato and E. Jose, Asymptotic behavior of the approximate controls for parabolic equations with interfacial contact resistance, ESAIM Control Optim. Calc. Var., 21 (2015), 138-164. doi: 10.1051/cocv/2014029. Google Scholar

[25]

P. Donato and E. Jose, Approximate controllability of a parabolic system with imperfect interfaces, Philipp. J. Sci., 144 (2015), 187-196. Google Scholar

[26]

P. DonatoK. H. Le Nguyen and R. Tardieu, The periodic unfolding method for a class of imperfect trasmission problems, J. Math. Sci., 176 (2011), 891-927. doi: 10.1007/s10958-011-0443-2. Google Scholar

[27]

P. Donato and K. H. Le Nguyen, Homogenization of diffusion problems with a nonlinear interfacial resistance, Nonlinear Differ. Equ. Appl., 22 (2015), 1345-1380. doi: 10.1007/s00030-015-0325-2. Google Scholar

[28]

P. Donato and S. Monsurrò, Homogenization of two heat conductors with interfacial contact resistance, Anal. Appl., 2 (2004), 247-273. doi: 10.1142/S0219530504000345. Google Scholar

[29]

P. DonatoS. Monsurrò and F. Raimondi, Homogenization of a class of singular elliptic problems in perforated domains, Nonlinear Anal., 173 (2018), 180-208. doi: 10.1016/j.na.2018.04.005. Google Scholar

[30]

P. Donato and A. Nabil, Approximate controllability of linear parabolic equations in perforated domains, ESAIM Control Optim. Calc. Var., 6 (2001), 21-38. doi: 10.1051/cocv:2001102. Google Scholar

[31]

T. DuranteL. Faella and C. Perugia, Homogenization and behaviour of optimal controls for the wave equation in domains with oscillating boudary, Nonlinear Differ. Equ. Appl., 14 (2007), 455-489. doi: 10.1007/s00030-007-3043-6. Google Scholar

[32]

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. Google Scholar

[33]

T. Durante and T. A. Mel'nyk, Homogenization of quasilinear optimal control problems involving a thick multilevel junction of type 3:2:1, ESAIM Control Optim. Calc. Var., 18 (2012), 583-610. doi: 10.1051/cocv/2011107. Google Scholar

[34]

L. Faella and S. Monsurrò, Memory effects arising in the homogenization of composites with inclusions, in Topics on Mathematics for Smart System doi: 10.1142/9789812706874_0008. Google Scholar

[35]

L. FaellaS. Monsurrò and C. Perugia, Homogenization of imperfect transmission problems: The case of weakly converging data, Differential Integral Equations, 31 (2018), 595-620. Google Scholar

[36]

L. FaellaS. Monsurrò and C. Perugia, Exact controllability for an imperfect transmission problem, J. Math. Pures Appl., 122 (2019), 235-271. doi: 10.1016/j.matpur.2017.11.011. Google Scholar

[37]

L. Faella and C. Perugia, Homogenization of a Ginzburg-Landau problem in a perforated domain with mixed boundary conditions, Bound. Value Probl., 2014 (2014), 28pp. doi: 10.1186/s13661-014-0223-2. Google Scholar

[38]

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. Google Scholar

[39]

L. Faella and C. Perugia, Optimal control for a hyperbolic problem in composites with imperfect interface: A memory effect, Evol. Equ. Control Theory, 6 (2017), 187-217. doi: 10.3934/eect.2017011. Google Scholar

[40]

E. Fernandez-Cara, Null controllability of the semilinear heat equation, ESAIM Control Optim. Calc. Var., 2 (1997), 87-103. doi: 10.1051/cocv:1997104. Google Scholar

[41]

H. C. Hummel, Homogenization for heat transfer in polycristals with interfacial resistances, Appl. Anal., 75 (2000), 403-424. doi: 10.1080/00036810008840857. Google Scholar

[42]

A. M. Khludnev, L. Faella and C. Perugia, Optimal control of rigidity parameters of thin inclusions in composite materials, Z. Angew. Math. Phys., 68 (2017), Art. 47, 12 pp. doi: 10.1007/s00033-017-0792-x. Google Scholar

[43]

L. Li and X. Zhang, Exact controllability for semilinear wave equations, J. Math. Anal. Appl., 250 (2000), 589-597. doi: 10.1006/jmaa.2000.6998. Google Scholar

[44]

J. L. Lions, Contrôlabilité Exacte et Homogénéisation, I, Asymptotic Anal., 1 (1988), 3-11. Google Scholar

[45]

J. L. Lions, Contrôlabilité Exacte, Stabilization at Perturbations De Systéms Distributé, Tomes 1, 2, Massonn, RMA, 829, 1988. Google Scholar

[46]

J. L. Lions and E. Magenes, Non-homogeneous Boundary Value Problems and Applications, Vol I, Springer-Verlag Berlin Heidelberg, New York, 1972. Google Scholar

[47]

R. Lipton, Heat conduction in fine scale mixtures with interfacial contact resistance, SIAM J. Appl. Math., 58 (1998), 55-72. doi: 10.1137/S0036139995295153. Google Scholar

[48]

R. Lipton and B. Vernescu, Composite with imperfect interface, Proc. R. Soc. Lond. Ser. A, 452 (1996), 329-358. doi: 10.1098/rspa.1996.0018. Google Scholar

[49]

S. Monsurrò, Homogenization of a two-component composite with interfacial thermal barrier, Adv. Math. Sci. Appl., 13 (2003), 43-63. Google Scholar

[50]

S. Monsurrò, Erratum for the paper "Homogenization of a two-component composite with interfacial thermal barrier", Adv. Math. Sci. Appl., 14 (2004), 375-377. Google Scholar

[51]

S. Monsurrò, Homogenization of a composite with very small inclusions and imperfect interface, in Multiscale problems and asymptotic analysis, GAKUTO Internat. Ser. Math. Sci. Appl., 24, Gakkotosho, Tokyo, (2006), 217–232. Google Scholar

[52]

T. Muthukumar and A. K. Nandakumaran, Homogenization of low-cost control problems on perforated domains, J. Math. Anal. Appl., 351 (2009), 29-42. doi: 10.1016/j.jmaa.2008.09.048. Google Scholar

[53]

A.K. NandakumaranR. Prakash and B. C. Sardar, Asymptotic analysis of Neumann periodic optimal boundary control problem, Math. Methods Appl. Sci., 39 (2016), 4354-4374. doi: 10.1002/mma.3865. Google Scholar

[54]

Z. Yang, Homogenization and correctors for the hyperbolic problems with imperfect interfaces via the periodic unfolding method, Commun. Pure Appl. Anal., 13 (2014), 249-272. doi: 10.3934/cpaa.2014.13.249. Google Scholar

[55]

Z. Yang, The periodic unfolding method for a class of parabolic problems with imperfect interfaces, ESAIM Math. Model. Numer. Anal., 48 (2014), 1279-1302. doi: 10.1051/m2an/2013139. Google Scholar

[56]

E. Zuazua, Exact boundary controllability for the semilinear wave equation, in Nonlinear Partial Differential Equations and Their Applications, Collège de France Seminar, Vol. X (Paris, 1987–1988), Pitman Res. Notes Math. Ser., 220, Longman Sci. Tech., Harlow, 1991, 357–391. Google Scholar

[57]

E. Zuazua, Approximate controllability for linear parabolic equations with rapidly oscillating coefficients, Control Cybernet., 23 (1994), 793-801. Google Scholar

[58]

E. Zuazua, Controllability of partial differential equations and its semi-discrete approximations, Discrete Contin. Dyn. Syst., 8 (2002), 469-513. doi: 10.3934/dcds.2002.8.469. Google Scholar

Figure 1.  The two-component domain $ \Omega $
Figure 2.  The sets $ \widehat{\Omega }^{\varepsilon } $ and $ \Lambda^\varepsilon $
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