A set-valued approach to control immunotherapy

In this paper, we use viability theory and set-valued analysis to investigate destroying cancer cells from a tissue that is treated by immunotherapy. The problem is set as a target control problem under state-control constraints, the mathematical model being a control system of three nonlinear ODEs. The goal of the therapy is to get a decreasing tumor cell density that attains zero at terminal time. We derive a family of continuous protocol laws and prove that the minimal selection of the feedback map, though it is discontinuous, stands for the best protocol law that involves minimum doses to clear tumor cells.