A geometric analysis of bang-bang extremals in optimal control problems for combination cancer chemotherapy

Cell-cycle specific compartmental models for the growth of cancer cells under combination chemotherapies are considered as multi-input optimal control problems over a fixed therapy interval. The controls are the dose rates of various chemotherapeutic agents, such as cytotoxic (killing) and cytostatic (blocking) drugs or recruiting agents. Singular controls are not optimal for the models under consideration and thus bang-bang controls become the natural candidates for optimality. We use a geometric approach based on the construction of a field of bang-bang extremals to determine the strong local optimality of extremals. If the flows of trajectories at a junction cross the switching surface transversally (transversal crossing), then local optimality is retained while it ceases if the two flows cross the switching surface in opposite directions (transversal folds). In the latter case, switching points are conjugate points for the combined flow. A simple algorithm will be described that allows us to verify if a junction is a transversal crossing or fold.