Approximation of Boundary Control Problems on Curved Domains. II - The Dirichlet Case
Résumé
The influence of small boundary variations of the domain on optimal controls is investigated in this paper. The domain variations are governed by a small parameter $h\to 0$. In a previous paper we have studied the Neuman control problem. In this paper, the Dirichlet control problem is considered. The optimal solutions are compared between the problems defined in the curved domain $\Omega$ and the polygonal domains $\Omega_h$, in the norm defined on the fixed boundary $\Gamma=\partial\Omega$ of the curved domain. To this end, an appropriate parametrization of the boundaries is introduced, and a one-to-one mapping between the boundaries $\Gamma$ and $\Gamma_h$ is employed. Error estimates of the order $h$ in the norm on the fixed boundary $\Gamma$ are derived for the difference of optimal controls.
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