APPROXIMATION OF BOUNDARY CONTROL PROBLEMS ON CURVED DOMAINS

摘要

In this paper we consider boundary control problems associated to a semilinear elliptic equation defined in a curved domain Omega. The Dirichlet and Neumann cases are analyzed. To deal with the numerical analysis of these problems, the approximation of Omega by an appropriate domain Omega(h) ( typically polygonal) is required. Here we do not consider the numerical approximation of the control problems. Instead, we formulate the corresponding infinite dimensional control problems in Omega(h), and we study the influence of the replacement of Omega by Omega(h) on the solutions of the control problems. Our goal is to compare the optimal controls defined on Gamma = partial derivative Omega with those defined on Gamma(h) = partial derivative Omega(h) and to derive some error estimates. The use of a convenient parametrization of the boundary is needed for such estimates.