reaches the line p = q. We will use these values to …nd the optimal control and trajectory. For
this valule of k = 6, it is easy to observe from Figure 6.2, that we have a very short period to
administer the full dose (u = 70) and we rapidly switch to partial dosage (singular control).
The partial dosage (u = usin) in this case will take the longer portion of the treatment.
Finally, we observe a short period of time, where the after e¤ect of the medication (u = 0)
carries on. This type of optimal control we will denote "as0"control. Figure 6.3 contains
the graph of the optimal trajectory which illustrates the response of the tumor and the
carrying capacity to such treatment. Note that very little shrinkage of tumor cells takes
place during the period of full dose while the decrease of volume of tumor cells observed
during the singular phase is considerably greater.
x 10
4
2
1.9
1.8
1.7
1.6
st
l co
a
1.5
fin
mi n i mu m
1.4
1.3
1.2
1.1
1
0 5 10 15 20 25 30 35 40
t i me
RFigure 6.1: Cost function for p(T) + 6 T
0 u(t)dt
38
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