Conclusion
In this paper, we analyzed the model for anti-angiogenesis proposed in [3] by Hahnfeldt
et al (Harvard School of Medecine) as an optimal control problem. The objective of the
problem was extended from the one considered in [6] where the �nal size of the tumor was
minimized by including the penalty on the cost of treatment.
It was shown how optimal control and corresponding optimal trajectory (tumor and
vascular response to the treatment) can be computed for this version of the problem. It
turnedoutthatthequalitativestructureofsolutionswillnotchangeunderthisnewobjective,
i.e., the typical trajectory will still follow the framework of "as0"(full dose, partial doses,
no dose). However, it was observed in the numerical simulations that for some values of
the parameter k (weight at the objective) k 2 (16:65;16:66), the behavior of the solutions
drastically changes. The singular arc stops being optimal and for some small interval of
time, the optimal controls are of type "a0". Then they hit another critical point after which
it is optimal not to administer the drug at all (optimal control is of type "0").
This brings us to an interesting conclusion that the extension in the objective by the
addition of the penalty on the cost of treatment does not change qualitatively the structure
of solutions whereas the choice of the weight in this penalty term does change the solution
qualitatively and in some cases even leads to trivial solutions. Thus, the values of this
parameter should be chosen carefully to properly re�ect the biomedical background of the
problem.
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