In this paper, we study the asymptotic behaviour of a parabolic optimal control problem in a domain Ωε ᴄ Rn, whose boundary ∂Ωε 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.
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