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In this example we choose another nonlinear partial differential
equation and homogeneous Dirichlet boundary conditions:
This problem has been considered in [16], but without state constraints.
Table 3:
Information on solution of Example 3
N+1 
it 
CPU 
Acc 

50 
29 
131 
8 
.110242 
100 
32 
2257 
8 
.110263 
200 
31 
42644 
8 
.110269 

Figure 5:
Example 3,
: Optimal control.

Figure 6:
Example 3,
: Optimal state and adjoint variable

The adjoint equations (2.10), (2.12) yield

(4.4) 
The minimum condition (2.21) leads again to the projection

(4.5) 
with
.
The optimal control is shown in Figure 5, while Figure 6 displays the
optimal state and associated adjoint variable.
The adjoint variable allows to verify the control rule (4.5).
Note that condition (2.7) does not hold for this example
in view of
.
The state is active in the points
.
In view of the decomposition (2.15), the measure is
given here by the singular measure
, where
are identified with the multipliers in view of
(3.15). Let us check now the accuracy with which the computed
adjoint equation is satisfied. At the point
we obtain
.
Then the discretized adjoint equation (3.11) yields
for .
Next: Example 4
Up: Numerical examples
Previous: Example 2
Hans D. Mittelmann
20001006