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## Discretization of the distributed control problem

The optimization variable in (1) is now taken as the vector

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 :
 (18)

The Dirichlet condition (28) is incorporated by fixing the values on :
 (19)

Observing the approximation (5) of the outward normal derivative, the discrete form of the Neumann boundary condition in (27) leads to the equality constraints
 (20)

The control and state inequality constraints (29) and (30) yield the inequality constraints
 (21) (22)

Note again that these inequality constraints do not depend on the meshsize . The discretized form of the cost function (25) is
 (23)

Hence, for distributed control problems the NLP-problem (1) is given by the relations (18)-(23). The corresponding Lagrangian function is
 (24)

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 state variables with indices we obtain the relations
 (25)

Here as in (12), the undefined multipliers are set to
 (26)

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
 (27)

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 ,
 (28)

while for a delta distribution we deduce from (27) the approximation
 (29)

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
 (30)

Next: Optimization codes and modeling Up: Discretization and optimization techniques Previous: Discretization of the boundary
Hans D. Mittelmann
2000-12-09