dimensions) of endothelial cells. Hence the inhibitor function is denoted by:
2I(p;q) = dp
3q (2.3)
where d is a constant that represents the death rate. The second result from [3] suggests
that the inhibitor function will grow at a rate qp faster than the stimulator function and
the relationship between both powers is such that  +  = 23:
In this model,  = 1 and  = 13: Thus the following formula represents the stimulator
function:
S(p;q) = bp: (2.4)
The …nal goal of anti-angiogenesis is to successfully eliminate the cancerous cells and to
shrink the tumor as much as possible while using the least amount of drug.
We are now ready to reformulate our model as an optimal control problem (OC). For a
free terminal time T, we want to minimize the value of
Z T
p(T) + k u(t)dt (2.5)
0
over all piecewise continuous (more generally Lebesque measurable functions)
u : [0;T] ! [0;a] (2.6)
subject to the following system of di¤erential equations
 

p = pln pq p(0) = p0
(2.7) 

2q = bp  + dp
3 q Guq q(0) = q0:
In [6] and [5], the model for anti-angiogenesis formulated here was considered with ob-
Rjective to minimize p(T) under the constraint T
0 u(t)dt  A (A 2 R is given). This means
7
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