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IMA Journal of Mathematical Control and Information Advance Access originally published online on October 9, 2008
IMA Journal of Mathematical Control and Information 2008 25(4):461-489; doi:10.1093/imamci/dnn010
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© The author 2008. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.

Gap phenomenon in the homogenization of parabolic optimal control problems

C. D'Apice

Department of Applied Mathematics, University of Salerno, Via Ponte don Melillo, 84084 Fisciano (SA), Italy

U. De Maio

Department of Applied Mathematics, University of Naple ‘FedericoII’ Complesso Monte S.Angelo, via Cintia, 80126 Napoli, Italy

P. I. Kogut{dagger}

Department of Differential Equations, Dnipropetrovsk National University, Naukova street, 13, DNU, 49050 Dnipropetrovsk, Ukraine

{dagger} Corresponding author. Email: p.kogut{at}i.ua

Received on February 4, 2007; Accepted on July 10, 2007

In this paper, we study the asymptotic behaviour of a parabolic optimal control problem in a domain {Omega}{epsilon} sub Formula , whose boundary {partial}{Omega}{epsilon} contains a highly oscillating part. We consider this problem with two different classes of Dirichlet boundary controls, and, as a result, we provide its asymptotic analysis with respect to the different topologies of homogenization. It is shown that the mathematical descriptions of the homogenized optimal control problems have different forms and these differences appear not only in the state equation and boundary conditions but also in the control constraints and the limit cost functional.

Keywords: optimal control problem; homogenization; thick multi-structure; variational convergence; set convergence; gap phenomenon.


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