Journal of the Ramanujan Mathematical Society

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Asymptotic Analysis of Optimal Controls of a Semilinear Problem in a Perforated Domain


Affiliations
1 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-8370459, Chile
2 Normandie Universite, Universite De Rouen, Laboratoire De Math Ematiques Raphel Salem, CNRS, UMR 6085, Avenue De l’Universite, BP 12, 76801 Saint-E´Tienne Du Rouvray Cedex, France
3 Institute of Mathematical Sciences and Physics, UP Los Banos, College, Los Banos, Laguna, Philippines
4 Department of Mathematics, Indian Institute of Science Education and Research Bhopal, Indore By-Pass Road, Bhauri, Bhopal-462066, India
     

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In this paper, we study the L2 and H1-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 H1-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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  • Asymptotic Analysis of Optimal Controls of a Semilinear Problem in a Perforated Domain

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Authors

Carlos Conca
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-8370459, Chile
Patrizia Donato
Normandie Universite, Universite De Rouen, Laboratoire De Math Ematiques Raphel Salem, CNRS, UMR 6085, Avenue De l’Universite, BP 12, 76801 Saint-E´Tienne Du Rouvray Cedex, France
Editha C. Jose
Institute of Mathematical Sciences and Physics, UP Los Banos, College, Los Banos, Laguna, Philippines
Indira Mishra
Department of Mathematics, Indian Institute of Science Education and Research Bhopal, Indore By-Pass Road, Bhauri, Bhopal-462066, India

Abstract


In this paper, we study the L2 and H1-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 H1-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.