Hence it suffices to work with a reduced number of optimization variables in (4.1), namely we can take

The

for all indices where the values on the boundary are substituted from the relations (4.15). The control and state inequality constraints (3.3) resp. (3.4) give rise to the

Finally, the cost function is derived from (3.1) as

By means of (4.16)-(4.19) we have obtained an NLP-problem of the form (4.1).

Let us evaluate the first order optimality conditions of Kuhn-Tucker type
for problem (4.16)-(4.19).
The Lagrangian function becomes

where the Lagrange multipliers , and

which are associated with the equality constraints (4.16), the inequality constraints (4.17), and the inequality constraints (4.18). The multipliers and satisfy the complementarity conditions corresponding to (3.9):

In the next step we discuss the necessary conditions of optimality

for different combinations of indices . For indices we obtain the relations

Hence, for this set of indices we see that the Lagrange multipliers satisfy the five-point-star difference equations for the adjoint equation in (3.6) if we use the same identification for the Borel measure as in (4.10),

from which we also get the special cases (4.11) and (4.12). The situation is different for indices with either or . E.g., for indices we get

Note that this equation does not involve the variable on the boundary. Defining we arrive again at the discretized form of the adjoint equation (3.6) using approximations as in (4.10)-(4.12). Similar relations hold for index combinations or or . For the remaining indices we find, e.g. for :

These relations do not involve the boundary variables and . Again, defining and we recover the adjoint equations. In summary, by requiring the Dirichlet boundary condition for the adjoint variables,

we see that the adjoint equation holds for all indices

The necessary conditions with respect to the control variables
are determined for the indices
by

Observing and the approximation of the normal derivative in (4.4), the minimum condition (3.8) holds with the identifications

Similar identifications hold for the other indices .

2002-11-25