Asymptotic behavior of the approximate controls for parabolic equations with interfacial contact resistance

Donato, Patrizia; Jose, Editha C.

ESAIM: Control, Optimisation and Calculus of Variations, Tome 21 (2015) no. 1, pp. 138-164.

In this paper, we study the approximate control for a class of parabolic equations with rapidly oscillating coefficients in an $\epsilon$-periodic composite with an interfacial contact resistance as well as its asymptotic behavior, as $\epsilon \to 0$. The condition on the interface depends on a parameter $\gamma \in (-1,1]$. The case $\gamma =1$ 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 $J_{\epsilon }$. The more interesting result of this study proves that the control and the corresponding solution of the $\epsilon$-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.

F. Ammar Khodja, Personal communication.

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