The First Order Optimality Conditions for the Discretized Problem

If we consider the discretized optimization problem as a finite dimensional in then the following optimality conditions arise (subscripts denote partial derivatives):

If one compares these optimality conditions with the optimiality conditions for the infinite dimensional problem , then one observes in a difference in the formula for the gradient. This difference is due to the fact that here we have neglected the infinite dimensional structure and computed the gradient with respect to the Euclidean scalar product in .

If we use the scalar product for the controls, which is the proper discretization of the scalar product in , then the gradient equation is given by

This formulation of the reduced gradient and the use of the weighted scalar product corresponds to the infinite dimensional formulation. We will see that this is crucial to compute meaningful controls.