Hence for a constant value v 2 [0;a]; and for q = 0;
bpq = f
v(p) = 2 : (3.30) + Gv + dp
3
The family of functions fv are all stricly increasing (see Figure 3.1) and bounded by fv(0) = 0
and fv(peq)(upper bound)
3b(b
2f d )
v(peq) = 3 2 (3.31) + Gv + d((b
d )2)3
3
1 )
2
= b d (b 2)Gv +  + (b (3.3)
3 b 1
2
= (b) (3.33)b + Gv d
b= p
b + Gv eq: (3.34)
On the following boundary segment, Q = f(p;q) : 0 < p < peq;q = qeqg;
  

bpq =
2
2q
eq Gu +  + dp3 < 0: (3.35) + dp
3 + Gu
The point (peq;qeq) is the equilibrium point for u = 0 and (p;q) 2 D: Hence, regardless of
the values taken by the control u, the trajectory always remains within the boundary of D.
However for the control taking constant value u = a, the equilibrium becomes:
3 bGa
2
peq = qeq = : (3.36)d
When b  Ga; the system no longer has an equilibrium point and now all the trajectories
converge to the origin as t goes to in…nity. This is a desired situation which means that
theoretically (i.e. given in…nite amount of time and inhibitors), we could eradicate the
13
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