As we start increasing the value of k; the graphs change in di¤erents ways. For the graph
of the optimal control, the time needed to give full dose (u = a) becomes longer; the after
e¤ect of the medication (u = 0) also takes a longer time while on the other hand, the portion
for the singular control becomes shorter (see Figure 6.5). The same behavior applies to
the graph of the corresponding optimal trajectory (see Figure 6.6). The change of behavior
of the graphs could be directly explained by the fact that the minimum point of the cost
function gets closer and closer to the origin as k is increasing (see Figure 6.4).
Figure 6.7 represents this cost for k = 15 whereas Figure 6.8 and Figure 6.9 are the
graphs for corresponding optimal control and trajectory. We also notice that optimal time
T is only about 1:8 and the singular portion of the optimal control is less than 1.4 (green
portion in Figure 6.8). Increasing further the value of k will bring us to a critical point where
the qualitative structure of the solutions will change.
6.1 Critical points
Firstly, there exist some values 1; 2 (16:65 < 1 < 2 < 16:66) such that, when k 2
(16:65; 1), we get an optimal control of type "as0". Secondly, there exist some values of
k; such that if k 2 ( 1; 2); the control obtained is of type "a0". Finally, if k 2 ( 2;16:66),
the optimal solution would be to adminster no drug (optimal control of type "0"). The
behavior of the graphs change drastically when k varies from 16:65 to 16:66. Even though
the values of k are relatively close to each other, the behavior observed in Figure 6.11 shows
that for k = 16:66, the function of cost has its minimum at the very …rst point of the curve.
Although, for k = 16:65; we can still see a minimum very close to the starting point (see
Figure 6.10). This means that our …nal time is T = 0 for k = 16:66; and it is better not
to administer any drug to the patient, whereas for k = 16:65, we still have a therapy of
the form as0 given in Figure 6.12. Figure 6.13 shows the comparison of optimal trajectories
for k = 16:65 and k = 16:66. Note that the blue trajectory (k = 16:65) still preserved the
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