Journal of the Ramanujan Mathematical Society

Volume 31, Issue 3, September 2016 pp. 265–305.

Asymptotic analysis of optimal controls of a semilinear problem in a perforated domain

Authors: Carlos Conca, Patrizia Donato, Editha C. Jose and Indira Mishra
Author institution:Department of Mathematical Engineering (DIM), Center for Mathematical Modelling (CMM, UMI CNRS 2807), Center for Biotechnology and Bioengineering (CeBiB), University of Chile, Casilla 170-3, Correo 3, Santiago 837 0459, Chile

Summary: In this paper, we study the L 2 and H 1-approximate controllability and homogenization of a semilinear elliptic boundary value problem in a perforated domain. The principal term in the state equation has rapidly oscillating coefficients and the control region is free from perforations (holes). The observable zone is locally distributed in the perforation free region, in the case of H 1-approximate controllability. By using the constructive approach introduced by Lions and which is based on the Fenchel--Rockafellar's duality theory, we obtain the approximate control of minimal norm. The existence of the control is established by means of a fixed point argument. Another interesting result of this study is that the minimal norm controls of the ε-problem converge to the optimal controls associated with the homogenized problem. The result in the case of rapidly oscillating coefficients in a fixed domain was proved in [Conca, et~al., J. Math. Anal. 285 (2003), 17--36]. The main difficulty relies in passing to the limit in the cost functional (as ε → 0) having rapidly oscillating coefficients.

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