Due to the Dirichlet conditions (28), the variables for can be eliminated from the optimization process.

The application of the five-point-star to the elliptic equation in (26) yields the following equations for all :

The Dirichlet condition (28) is incorporated by fixing the values on :

Observing the approximation (5) of the outward normal derivative, the discrete form of the Neumann boundary condition in (27) leads to the

The control and state inequality constraints (29) and (30) yield the

Note again that these inequality constraints do not depend on the meshsize . The discretized form of the cost function (25) is

Hence, for distributed control problems the NLP-problem (1) is given by the relations (18)-(23). The corresponding Lagrangian function is

where the Lagrange multipliers , and are associated with the equality constraints (18) and (20), resp. the inequality constraints (21) and (22). The multipliers and satisfy complementarity conditions corresponding to (37):

The discussion of the necessary conditions of optimality

is similar to that for boundary control problems. For

Here as in (12), the undefined multipliers are set to

in accordance with the Dirichlet condition (35). We deduce from equations (25) that the Lagrange multipliers satisfy the five-point-star difference equations for the adjoint equation in (33) if we approximate the measure and the multiplier function in by

where denotes a square centered at with area . Recall again the decomposition (38) of the measure . If the singular part of the measure vanishes, then (27) yields an approximation for the density ,

while for a delta distribution we deduce from (27) the approximation

For indices
on the boundary
we obtain, e.g., for
,

which constitutes the discrete version of the Neumann boundary condition (34).

Finally, necessary conditions with respect to the *control variables*
for
are determined by

>From this equation we can recover the discrete version of the control law (12), if we use the identification

2000-12-09