Bifurcation of singular arcs in an optimal control problem for cancer immune system interactions under treatment

A mathematical model for cancer treatment that includes immunological activity is considered as an optimal control problem. In the uncontrolled system there exist both a region of benign and of malignant cancer growth separated by the stable manifold of a saddle point. The aim of treatment is to move an initial condition that lies in the malignant region into the region of benign growth. In our formulation of the objective a penalty term is included that approximates this separatrix by its tangent space and minimization of the objective is tantamount to moving the state of the system across this boundary. In this paper, for various values of a parameter that describes the relative effectiveness of the killing action of a cytotoxic drug on cancer cells and the immunocompetent cell density, the existence and optimality of a singular arc is analyzed. The question of existence of optimal controls and the structure of near-optimal protocols that move the system into the region of attraction of the benign stable equilibrium is discussed.