Finite Element Approximation of Elliptic Dirichlet Optimal Control Problems

In this paper, we present a priori error analysis for the finite element discretization of elliptic optimal control problems, where a finite dimensional control variable enters the Dirichlet boundary conditions. The analysis of finite element approximations of optimization problems governed by partial differential equations is an area of active research, see, e.g., [1, 12, 17, 18]. The consideration of Dirichlet boundary control problems is more difficult than Neumann control or control by right-hand side, from both the theoretical and the numerical point of view, because the Dirichlet boundary conditions do not directly enter the variational setting. Moreover, the necessary optimality conditions described by an optimality system usually contain the normal derivative of the adjoint state, see, e.g., [16] or [8]. This fact complicates the achievement of the optimal order of convergence in the context of finite element discretization.