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Discretization of the boundary control problem

Let denote approximations of the state variables for and let be approximations for the control variables for . We specify the functions for the optimization problem (1) corresponding to problem (1)-(6) as follows. The optimization variable in (1) is taken as the vector

Note that we do not consider the variables explicitly as optimization variables since they are prescribed by the Dirichlet condition (4).

Equality constraints are obtained by applying the five-point-star to the elliptic equation in (2) in all points with ,

 (3)

In these equations may appear the undefined variables for . These variables have to be substituted by the Dirichlet conditions (4),
 (4)

The derivative in the direction of the outward normal is approximated by the expression where
 (5)

Then the discrete form of the Neumann boundary condition (3) leads to the equality constraints
 (6)

The control and state inequality constraints (5) and (6) yield the inequality constraints
 (7) (8)

Observe that the inequality constraints do not depend on the meshsize . Later on, this fact will require a scaling of the Lagrange multipliers. Finally, the discretized form of the cost function (1) is
 (9)

Then the relations (2)-(8) define an NLP-problem of the form (1).

Associate Lagrange multipliers , and with the equality constraints (3) and (6) resp. the inequality constraints (7) and (8). Then the Lagrangian function for the above NLP-problem becomes:

 (10)

The multipliers and satisfy complementarity conditions corresponding to (14):

Now we discuss the necessary conditions of optimality

for state and control variables assuming different combinations of indices . For state variables with indices we obtain the relations
 (11)

These equations contain multipliers for that do not appear in the Lagrangian (10). To make equations and definitions consistent, we put
 (12)

This substitution corresponds to the Dirichlet condition (11). Relations (11) then reveal that the Lagrange multipliers satisfy the five-point-star difference equations for the adjoint equation in (9) if we use the following approximation for the Borel measure ,
 (13)

where denotes a square centered at with area . Recall the decomposition (15) of the measure . If the singular part of the measure vanishes, i.e. then (13) yields the following appro-
ximation for the density ,
 (14)

In case that the measure is a delta distribution, we obtain from (13) the relation
 (15)

On the boundary part we get for indices assuming, e.g., :

Recalling (5) this represents the discrete version of the Neumann boundary condition (10).

Finally, necessary conditions with respect to the control variables for indices are determined by the following two relations. For with, e.g., we get

This equation yields the discrete version of the optimality condition (12) for the control, if we use the identification
 (16)

For indices with, e.g., we find

Observing and the approximation (5) of the normal derivative, the minimum condition (13) holds with the substitutions
 (17)

Next: Discretization of the distributed Up: Discretization and optimization techniques Previous: Discretization and optimization techniques
Hans D. Mittelmann
2000-12-09