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
(subscripts denote partial derivatives):
If one compares these optimality conditions
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.