Optimal Control for Anticancer Therapy

Abstract

In this report, the controlled Lotka–Volterra competition model is used to describe the interaction of the concentrations of healthy and cancer cells. For this controlled model, the minimization problem of the terminal functional is considered, which is a weighted difference of the concentrations of cancerous and healthy cells at the final moment of the treatment period. To analyze the optimal solution of this problem, which consists of the optimal control and the corresponding optimal solutions of the differential equations that determine the model, the Pontryagin maximum principle is applied. It allows to highlight the values of the model parameters under which the optimal control corresponding to them is a piecewise-constant function with at most one switching. Also, the values of the model parameters are found, under which the corresponding optimal control is either a bang–bang function with a finite number of switchings, or in addition to the bang–bang-type portions (nonsingular portions), it also contains a singular arc. Further, only numerical investigations of the optimal control are possible. Therefore, the report presents the results of numerical calculations performed using the software BOCOP-2.1.0 that lead us to the conclusions about the possible type of the optimal control and the corresponding optimal solutions.