Chapter 6
Numerical Simulations
For simulations purposes in this chapter, we will use mm3 for our measurements of p and
q volumes. Also, = 0:084=day, b = 5:85=day, d = 0:00873=mm3=day, G = 0:15kg=mg
of dose/day and �nally = 0:02=day ([3]): The initial volume of tumor cells used here is
p0 = 13;000, the initial value of the carrying capacity of the vasculature is q0 = 10;000; and
full dose is a = 70: Our purpose here is to minimize
Z T
p(T) + k u(t)dt: (6.1)
0
For several values of k, we will �nd the the �nal cost of control, optimal trajectory, and
optimal control used. For di�erent values of k, the graphs obtained will be relatively di�erent
from each other. We will then analyse the behavior of the graphs and provide explanations
for the di�erences observed.
We will start our simulation with smaller values of k and we will increment and observe
di�erent behaviors. For a starting value of k = 6; the graphs in Figures 6.3, 6.2, and 6.1
respectively, represent the optimal control, the optimal trajectory, and the cost function
R(f(u) = p(T) + k T
0 u(t)dt)); that we are minimizing. Figure 6.1 clearly shows that the
minimum of the cost in obtained for T 14: This is the time when the optimal trajectory
leaves the singular arc and the minimum value of the cost is realized when the trajectory
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