Approximation of Boundary Control Problems on Curved Domains. I - The Neumann Case

In this paper we consider a Neumann control problem associated to a semilinear elliptic equation defined in a curved domain $\Omega$. To deal with the numerical analysis of this problem, the approximation of $\Omega$ by an appropriate domain $\Omega_h$ (typically polygonal) is required. Then the same infinite dimensional control problem is formulated in $\Omega_h$. We study the influence of the replacement of $\Omega$ by $\Omega_h$ on the solutions of the control problem. Our goal is to compare the optimal controls defined on $\Gamma = \partial\Omega$ with those defined on $\Gamma_h = \partial\Omega_h$ and to derive some error estimates. The use of a convenient parametrization of the boundary is needed for such estimates