American Institute of Mathematical Sciences

Advanced
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 & 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.  Google Scholar

[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.   Google Scholar

[3]

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

[4]

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

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[25]

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

[26]

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

[27]

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

[28]

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

[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.  Google Scholar

[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.  Google Scholar

[31]

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

[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.   Google Scholar

[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.   Google Scholar

[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. Google Scholar

[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.  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]

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

[3]

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

[4]

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

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[25]

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

[26]

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

[27]

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

[28]

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

[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.  Google Scholar

[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.  Google Scholar

[31]

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

[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.   Google Scholar

[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.   Google Scholar

[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. Google Scholar

[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.  Google Scholar

Figure 1.  $\Omega_{\varepsilon}$
Figure Options

Download full-size image

Download as PowerPoint slide

[1]

Leszek Gasiński, Nikolaos S. Papageorgiou. Relaxation of optimal control problems driven by nonlinear evolution equations. Evolution Equations & Control Theory, 2020  doi: 10.3934/eect.2020050
[2]

Erik Kropat. Homogenization of optimal control problems on curvilinear networks with a periodic microstructure --Results on $\boldsymbol{S}$-homogenization and $\boldsymbol{Γ}$-convergence. Numerical Algebra, Control & Optimization, 2017, 7 (1) : 51-76. doi: 10.3934/naco.2017003
[3]

Kim Dang Phung, Gengsheng Wang, Xu Zhang. On the existence of time optimal controls for linear evolution equations. Discrete & Continuous Dynamical Systems - B, 2007, 8 (4) : 925-941. doi: 10.3934/dcdsb.2007.8.925
[4]

Ulisse Stefanelli, Daniel Wachsmuth, Gerd Wachsmuth. Optimal control of a rate-independent evolution equation via viscous regularization. Discrete & Continuous Dynamical Systems - S, 2017, 10 (6) : 1467-1485. doi: 10.3934/dcdss.2017076
[5]

Gengsheng Wang, Guojie Zheng. The optimal control to restore the periodic property of a linear evolution system with small perturbation. Discrete & Continuous Dynamical Systems - B, 2010, 14 (4) : 1621-1639. doi: 10.3934/dcdsb.2010.14.1621
[6]

Elimhan N. Mahmudov. Optimal control of evolution differential inclusions with polynomial linear differential operators. Evolution Equations & Control Theory, 2019, 8 (3) : 603-619. doi: 10.3934/eect.2019028
[7]

Leszek Gasiński. Optimal control problem of Bolza-type for evolution hemivariational inequality. Conference Publications, 2003, 2003 (Special) : 320-326. doi: 10.3934/proc.2003.2003.320
[8]

Norbert Koksch, Stefan Siegmund. Feedback control via inertial manifolds for nonautonomous evolution equations. Communications on Pure & Applied Analysis, 2011, 10 (3) : 917-936. doi: 10.3934/cpaa.2011.10.917
[9]

Urszula Ledzewicz, Stanislaw Walczak. Optimal control of systems governed by some elliptic equations. Discrete & Continuous Dynamical Systems - A, 1999, 5 (2) : 279-290. doi: 10.3934/dcds.1999.5.279
[10]

Eduardo Casas, Konstantinos Chrysafinos. Analysis and optimal control of some quasilinear parabolic equations. Mathematical Control & Related Fields, 2018, 8 (3&4) : 607-623. doi: 10.3934/mcrf.2018025
[11]

Elimhan N. Mahmudov. Infimal convolution and duality in convex optimal control problems with second order evolution differential inclusions. Evolution Equations & Control Theory, 2020  doi: 10.3934/eect.2020051
[12]

Elimhan N. Mahmudov. Optimal control of Sturm-Liouville type evolution differential inclusions with endpoint constraints. Journal of Industrial & Management Optimization, 2020, 16 (5) : 2503-2520. doi: 10.3934/jimo.2019066
[13]

Y. Efendiev, B. Popov. On homogenization of nonlinear hyperbolic equations. Communications on Pure & Applied Analysis, 2005, 4 (2) : 295-309. doi: 10.3934/cpaa.2005.4.295
[14]

Hakima Bessaih, Yalchin Efendiev, Florin Maris. Homogenization of the evolution Stokes equation in a perforated domain with a stochastic Fourier boundary condition. Networks & Heterogeneous Media, 2015, 10 (2) : 343-367. doi: 10.3934/nhm.2015.10.343
[15]

Gabriella Zecca. An optimal control problem for some nonlinear elliptic equations with unbounded coefficients. Discrete & Continuous Dynamical Systems - B, 2019, 24 (3) : 1393-1409. doi: 10.3934/dcdsb.2019021
[16]

Lucas Bonifacius, Ira Neitzel. Second order optimality conditions for optimal control of quasilinear parabolic equations. Mathematical Control & Related Fields, 2018, 8 (1) : 1-34. doi: 10.3934/mcrf.2018001
[17]

Peng Zhong, Suzanne Lenhart. Study on the order of events in optimal control of a harvesting problem modeled by integrodifference equations. Evolution Equations & Control Theory, 2013, 2 (4) : 749-769. doi: 10.3934/eect.2013.2.749
[18]

Ping Lin, Weihan Wang. Optimal control problems for some ordinary differential equations with behavior of blowup or quenching. Mathematical Control & Related Fields, 2018, 8 (3&4) : 809-828. doi: 10.3934/mcrf.2018036
[19]

Fulvia Confortola, Elisa Mastrogiacomo. Feedback optimal control for stochastic Volterra equations with completely monotone kernels. Mathematical Control & Related Fields, 2015, 5 (2) : 191-235. doi: 10.3934/mcrf.2015.5.191
[20]

Shu Luan. On the existence of optimal control for semilinear elliptic equations with nonlinear neumann boundary conditions. Mathematical Control & Related Fields, 2017, 7 (3) : 493-506. doi: 10.3934/mcrf.2017018

2019 Impact Factor: 0.953

Tools

Metrics

  • PDF downloads (38)
  • HTML views (47)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences