Analysis of a class of optimal control problems arising in cancer chemotherapy

Abstract

A class of mathematical models for cancer chemotherapy take the form of an optimal control problem over a fixed horizon with dynamics given by a bilinear system and objective linear in the control. In this paper we give results on local optimality of controls for both a two- and three-dimensional model. The main control in both models is a killing agent which is active during cell-division. The three-dimensional model also considers a blocking agent which slows down the growth of the cells during synthesis. The cumulative effect of the killing agent is used to model the negative effect of the treatment on healthy cells. It is shown that singular controls are not optimal for these models and optimality properties of bang-bang controls are established. Specifically, transversality conditions at the switching surfaces are derived which in a nondegenerate setting guarantee the local optimality of the flow if satisfied while they eliminate optimality of the trajectories if violated.