Sufficient conditions for optimality of controls in biomedical systems

A general class of optimal control problems of Bolza type is considered which arise in mathematical models of biomedical systems when chemotherapy treatment protocols for a disease over a fixed interval are considered. The controls represent the dosages of drugs administered and in this paper an objective which is quadratic in the controls is analyzed. This choice leads to continuous optimal controls which alternate between values in the interior and on the boundary of the control set. Using the method of characteristics a local field of extremals is constructed around a reference trajectory and sufficient conditions for strong local minima are given. These conditions apply to several biomedical models of chemotherapy for diseases like cancer or HIV infections which have been considered in the literature before, but not in the context of sufficient conditions. The result is illustrated with a simulation of a three-compartment model for cancer chemotherapy.