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June  2017, 6(2): 187-217. doi: 10.3934/eect.2017011

Optimal control for a hyperbolic problem in composites with imperfect interface: A memory effect

Luisa Faella 1, and  Carmen Perugia 2,,

1. 

Dipartimento di Ingegneria Elettrica e dell' Informazione, Università degli Studi di Cassino e del Lazio Meridionale, via G. Di Biasio 43, Cassino, 03043, Italia
2. 

Dipartimento di Scienze e Tecnologie, Università del Sannio, Via Port' Arsa 11, Benevento, 82100, Italia

* Corresponding author: Carmen Perugia

Received  September 2016 Revised  January 2017 Published  April 2017

Fund Project: Author's contributions: The authors conceived and wrote this article in collaboration and with same responsibility. All of them read and approved the final manuscript.

We study an optimal control problem for certain evolution equations in two component composites with $\varepsilon$-periodic disconnected inclusions of size $\varepsilon$ in presence of a jump of the solution on the interface that varies according to a parameter $γ$. In particular the case $γ=1$ is examinated.

Keywords: Homogenization, optimal control, evolution equations.
Mathematics Subject Classification: 35B27, 35L10, 49J20, 93C20.
Citation: Luisa Faella, Carmen Perugia. Optimal control for a hyperbolic problem in composites with imperfect interface: A memory effect. Evolution Equations and Control Theory, 2017, 6 (2) : 187-217. doi: 10.3934/eect.2017011
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.

[2]

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

[3]

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

[4]

D. Cioranescu and P. Donato, An Introduction to Homogenization Oxford Lecture Ser. Math., Appl., 17, Oxford University Press, New York, 1999.

[5]

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. 

[6]

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

[7]

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.

[8]

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.

[9]

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.

[10]

U. De Maio, A. 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.

[11]

P. Donato, Some corrector results for composites with imperfect interface, Rend. Mat. Appl(7), 26 (2006), 189-209. 

[12]

P. Donato, Homogenization of a class of imperfect transmission problems, in Multiscale Problems: Theory, Numerical Approximation and Applications, Series in Contemporary Applied Mathematics, A. Damlamian, B. Miara and T. Li Editors, Higher Education Press, Beijing, 16 (2011), 109–147. doi: 10.1142/9789814366892_0004.

[13]

P. Donato, L. 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.

[14]

P. Donato, L. 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.

[15]

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.

[16]

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

[17]

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

[18]

P. Donato and A. Nabil, Homogenization and correctors for the heat equation in perforated domains, Ricerche Mat., 50 (2001), 115-144. 

[19]

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.

[20]

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.

[21]

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, Optimisation and Calculus of Variations, 18 (2012), 583-610.  doi: 10.1051/cocv/2011107.

[22]

L. Faella and S. Monsurr`o, Memory effects arising in the homogenization of composites with inclusions, Topics on Mathematics for Smart System. World Sci. Publ., Hackensack, USA, (2007), 107–121. doi: 10.1142/9789812706874_0008.

[23]

L. Faella and C. Perugia, Homogenization of a Ginzburg-Landau problem in a perforated domain with mixed boundary conditions, Bound. Value Probl, 223 (2014), 1-28.  doi: 10.1186/s13661-014-0223-2.

[24]

L. Faella and C. Perugia, Optimal control for evolutionary imperfect transmission problems, Bound. Value Probl, 50 (2015), 1-16.  doi: 10.1186/s13661-015-0310-z.

[25]

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

[26]

J. L. Lions, Optimal Control of Systems Governed by Partial Differential Equations Springer-Verlag Berlin Heidelberg New York, 1971.

[27]

J. L. Lions, Contràlabilité Exacte et Homogénéisation, I. Asymptotic Analysis, 1 (1988), 3-11. 

[28]

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

[29]

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

[30]

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.

[31]

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

[32]

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

[33]

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

[34]

L. Tartar, Cours Peccot, Collège de France (March 1977). Partially written in F. Murat, HConvergence, Séminaire d'analyse fonctionnelle et numérique de l'Université d'Alger (1977-78). English translation in Mathematical Modeling of Composite Materials, A. Cherkaev and R. V. Kohon ed., Progress in Nonlinear Differential Equations and their Applications, Birkh¨auser-Verlag, (1997), 21–44.

[35]

E. Zeidler, Nonlinear Functional Analysis and its Applications, Vol Ⅱ, Part A and B Springer-Verlag, New York, 1986. doi: 10.1007/978-1-4612-4838-5.

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.

[2]

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

[3]

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

[4]

D. Cioranescu and P. Donato, An Introduction to Homogenization Oxford Lecture Ser. Math., Appl., 17, Oxford University Press, New York, 1999.

[5]

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. 

[6]

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

[7]

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.

[8]

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.

[9]

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.

[10]

U. De Maio, A. 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.

[11]

P. Donato, Some corrector results for composites with imperfect interface, Rend. Mat. Appl(7), 26 (2006), 189-209. 

[12]

P. Donato, Homogenization of a class of imperfect transmission problems, in Multiscale Problems: Theory, Numerical Approximation and Applications, Series in Contemporary Applied Mathematics, A. Damlamian, B. Miara and T. Li Editors, Higher Education Press, Beijing, 16 (2011), 109–147. doi: 10.1142/9789814366892_0004.

[13]

P. Donato, L. 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.

[14]

P. Donato, L. 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.

[15]

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.

[16]

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

[17]

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

[18]

P. Donato and A. Nabil, Homogenization and correctors for the heat equation in perforated domains, Ricerche Mat., 50 (2001), 115-144. 

[19]

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.

[20]

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.

[21]

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, Optimisation and Calculus of Variations, 18 (2012), 583-610.  doi: 10.1051/cocv/2011107.

[22]

L. Faella and S. Monsurr`o, Memory effects arising in the homogenization of composites with inclusions, Topics on Mathematics for Smart System. World Sci. Publ., Hackensack, USA, (2007), 107–121. doi: 10.1142/9789812706874_0008.

[23]

L. Faella and C. Perugia, Homogenization of a Ginzburg-Landau problem in a perforated domain with mixed boundary conditions, Bound. Value Probl, 223 (2014), 1-28.  doi: 10.1186/s13661-014-0223-2.

[24]

L. Faella and C. Perugia, Optimal control for evolutionary imperfect transmission problems, Bound. Value Probl, 50 (2015), 1-16.  doi: 10.1186/s13661-015-0310-z.

[25]

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

[26]

J. L. Lions, Optimal Control of Systems Governed by Partial Differential Equations Springer-Verlag Berlin Heidelberg New York, 1971.

[27]

J. L. Lions, Contràlabilité Exacte et Homogénéisation, I. Asymptotic Analysis, 1 (1988), 3-11. 

[28]

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

[29]

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

[30]

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.

[31]

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

[32]

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

[33]

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

[34]

L. Tartar, Cours Peccot, Collège de France (March 1977). Partially written in F. Murat, HConvergence, Séminaire d'analyse fonctionnelle et numérique de l'Université d'Alger (1977-78). English translation in Mathematical Modeling of Composite Materials, A. Cherkaev and R. V. Kohon ed., Progress in Nonlinear Differential Equations and their Applications, Birkh¨auser-Verlag, (1997), 21–44.

[35]

E. Zeidler, Nonlinear Functional Analysis and its Applications, Vol Ⅱ, Part A and B Springer-Verlag, New York, 1986. doi: 10.1007/978-1-4612-4838-5.

Figure 1.  $\Omega_{\varepsilon}$
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