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In this section we consider an optimal control problem for a semilinear elliptic equation of logistic type which was studied in Leung, Stojanovic
[21,28]. The problem is to determine a distributed control
that minimizes the functional

(4.7) 
subject to the elliptic state equation

(4.8) 
homogeneous Neumann boundary conditions,



(4.9) 
and
control and state inequality constraints



(4.10) 
Figure 11:
Example 6 : Optimal control.

Figure 12:
Example 6 : Optimal state and adjoint variable

Here, denotes
the population of a biological species, a spatially dependent
intrinsic growth rate, the crowding effect, while denotes the
difference between economic cost and revenue,
with nonnegative constants .
In [21,28] the function was chosen as .
Numerical results for this case can be found in [25].
Here, we consider
if
or and
otherwise.
The goal is to
find a control function which maximizes profit.
A similar control problem with Dirichlet boundary conditions was recently
studied by Cañada et al. [8].
The adjoint equations (2.10), (2.11) yield the following
equations:
For , the minimum condition (2.18) gives
the control law

(4.11) 
where
denotes the projection operator on the interval
.
The following concrete data were used:
Figure 11 displays the optimal control while Figure 12 shows
the optimal state and adjoint variable.
The reader may verify that the minimum condition given
by the projection (4.11) holds with high accuracy.
However, note that condition (2.7) imposed in [6]
is not satisfied everywhere in .
Table 7:
Information on solution of Example 6
N+1 
it 
CPU 
Acc 

50 
29 
104 
8 
4.19322 
100 
32 
2235 
8 
4.27569 
200 
33 
42543 
8 
4.31709 

Next: Conclusion
Up: Numerical examples
Previous: Example 5
Hans D. Mittelmann
20001006