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In this example we choose a nonlinear partial differential
equation and homogeneous Dirichlet boundary conditions:
These equations are related to a simplified GinzburgLandau model
for supercon ductivity in the absence of internal magnetic fields
with the wave function;
cf. Ito, Kunisch [19] and Kunisch, Volkwein [20].
The adjoint equations (2.10), (2.12) apply:

(4.1) 
Since , the minimum condition (2.21) yields the
control law

(4.2) 
where
denotes the projection operator onto the interval
.
The optimal control is shown in Figure 1, while Figure 2 depicts the
optimal state and associated adjoint variable.
The adjoint variable permits the verification of
the control law (4.2).
The state is active in the center .
Figure 2 indicates that condition (2.7) is not satisfied since
holds for all .
However, wellposedness of the GinzburgLandau model follows from
results in Gunzburger et al. [15].
Figure 1:
Example 1
: Optimal control.

Figure 2:
Example 1,
: Optimal state and adjoint variable

Table 1:
Information on solution of Example 1
N+1 
it 
CPU 
Acc 

50 
26 
104 
8 
.0577903 
100 
30 
1897 
8 
.0621615 
200 
35 
54831 
8 
.0644259 

Figure 3:
Example 2, : Optimal control and switching curve
.

Figure 4:
Example 2, : Optimal state and adjoint variable

Next: Example 2
Up: Numerical examples
Previous: Numerical examples
Hans D. Mittelmann
20001006