Heinz Schättler

On optimal controls for a general mathematical model for chemotherapy of HIV

We describe a class of optimal control problems which arise as mathematical models for biological systems in the chemotherapy of diseases which have a strong cell proliferation aspect such as cancer or AIDS. Although individually these problems are very different in their specifics, yet due to. the underlying mechanisms of cell dynamics, they also have many aspects in common and can be put into one general abstract mathematical model which encompasses them all. While on one side there is a need to consider-these problems individually to gain insight into implications for the underlying disease, on the other side there are also simplifications and insights to be gained by looking at the general properties common to all these models. In this paper we will develop and analyze such a structure in models for HIV-infection and anti-viral treatment of AIDS which have been proposed in the literature. Specifically, we give general sufficient conditions for strong local optimality of reference trajectories.

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

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.