The problem of determining optimal distributed and boundary controls for a distributed parameter system modelled by a non-linear partial differential equation defined of function spaces is considered. The performance index includes penalties on the state and its spatial derivatives throughout the interior and at the boundary of the spatial domain as well as penalties on the distributed and boundary controls. Under certain differentiability and well posedness assumptions the calculus of variations and an orthogonal representation of Green's identity are used to establish first and second-order necessary conditions on the optimal controls. The theory is then applied to a linear system with quadratic cost on the state and its spatial derivatives, and controls. Two specific examples of the linear system are considered and a method of decoupling the canonical equation for each example is presented.