The eigenvalues of J will be:
pB

1 = (3.25)2
q
( + b) ( + b)2 + 83 (b)
= < 0 (3.26)2
q
( + b) + ( + b)2 + 83 (b)
2 = (3.27)2
q
( + b) + ( + b)2
< = 0: (3.28)2
It is easily seen that the signs of 1 and 2 are negative. Therefore, our equilibrium for the
uncontrolled system is an asymptotically stable node.
Remark 2 The values of peq and qeq at the equilibrium are way too high to be acceptable
for medical relevancy. Therefore, it makes sense to restrict the values of p and q in the
following domain:
D = f(p;q) : 0 < p  peq;0 < q  qeqg: (3.29)
Figure 3.2 gives the phase portrait for the uncontrolled system (u = 0) for our parameter
values speci�ed earlier. Note that in this case, the equilibrium marked as * on the �gure takes
a value of over 16000 in cancer and endothelial cells, which medically is a very high value.
Proposition 3 For any admissible control u and arbitrary positive initial conditions p0 and
q0; the domain D (3.29) is positively invariant. D being positively invariant means that if
(p0;q0) 2 D; then for any admissible control u de�ned over the interval [0;1); the solution
(p(t);q(t)) for the dynamics with initial conditions (p0;q0) exits for all t  0 and (p(t);q(t)) 2
D:
Proof. The positive invariance of Q = f(p;q) : 0 < p; 0 < qg can be easily proved
with the results obtained in [4] for any admissible control u. In the boundary segment
P = f(p;q) : p = peq;0 < q < qeqg; it is also easy to see from (2.1) that p > q ) p_ < 0:
12
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