Volume 21, Number 1, January-March 2015
|Page(s)||138 - 164|
|Published online||03 December 2014|
Asymptotic behavior of the approximate controls for parabolic equations with interfacial contact resistance∗
Normandie Université, Université de Rouen, Laboratoire de
Mathématiques Raphaël Salem, CNRS, UMR 6085, Avenue de l’Université,
Saint-Étienne du Rouvray cedex,
2 Institute of Mathematical Sciences and Physics, UP Los Baños, College, Los Baños, Laguna, Philippines
Received: 29 August 2013
Revised: 8 May 2014
In this paper, we study the approximate control for a class of parabolic equations with rapidly oscillating coefficients in an ε-periodic composite with an interfacial contact resistance as well as its asymptotic behavior, as ε → 0. The condition on the interface depends on a parameter γ ∈ (−1,1]. The case γ = 1 is the most interesting one, and the more delicate, since the homogenized problem is given by coupled system of a P.D.E. and an O.D.E., giving rise to a memory effect. The variational approach to approximate controllability introduced by Lions in [J.-L. Lions. In Proc. of Jornadas Hispano-Francesas sobre Control de Sistemas Distribuidos, octubre 1990. Grupo de Análisis Matemático Aplicado de la University of Malaga, Spain (1991) 77–87] lead us to the construction of the control as the solution of a related transposed problem. The final data of this problem is the unique minimum point of a suitable functional Jε. The more interesting result of this study proves that the control and the corresponding solution of the ε-problem converge respectively to a control of the homogenized problem and to the corresponding solution. The main difficulties here are to find the appropriate limit functionals for the control of the homogenized system and to identify the limit of the controls.
Mathematics Subject Classification: 35B27 / 35Q93 / 93B05
Key words: Approximate controllability of parabolic equation / homogenization in a two-component domain with a periodic interface / jump condition on the interface depending on a parameter
© EDP Sciences, SMAI, 2014
Current usage metrics show cumulative count of Article Views (full-text article views including HTML views, PDF and ePub downloads, according to the available data) and Abstracts Views on Vision4Press platform.
Data correspond to usage on the plateform after 2015. The current usage metrics is available 48-96 hours after online publication and is updated daily on week days.
Initial download of the metrics may take a while.