Homogenization and exact controllability for problems with imperfect interface

Sara Monsurrò; Carmen Perugia; Sara Monsurrò; Carmen Perugia

doi:10.3934/nhm.2019017

Networks and Heterogeneous Media

2019, Volume 14, Issue 2: 411-444. doi: 10.3934/nhm.2019017

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Homogenization and exact controllability for problems with imperfect interface

Sara Monsurrò ^{1
,
,},
Carmen Perugia ^{2
,}

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

Received: 01 November 2018
35B27, 35J25, 35Q93, 82B24, 93B05

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.
- Homogenization,
- imperfect interface,
- jump boundary condition,
- weakly converging data,
- elliptic equations,
- exact controllability,
- evolution equations
Citation: Sara Monsurrò, Carmen Perugia. Homogenization and exact controllability for problems with imperfect interface[J]. Networks and Heterogeneous Media, 2019, 14(2): 411-444. doi: 10.3934/nhm.2019017

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Abstract

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.

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