Abstract
In this paper, an optimal boundary control problem for a distributed elliptic system on Lipschitz domains with boundary homogeneous Dirichlet conditions and independently with Neumann conditions is analysed. The necessary and sufficient optimality conditions for such problems with the quadratic cost functionals are obtained. A Jacobi spectral Galerkin method is introduced to develop a direct solution technique for the numerical solution of elliptic problems subject to Dirichlet and Neumann conditions in one and two dimensions. The numerical examples are included to demonstrate the validity and applicability of the techniques and comparison is made with the existing results. The method is easy to implement and yields very accurate results.