On Applications of the Hamilton—Jacobi Equations and Optimal Control Theory to Problems of Chemotherapy of Malignant Tumors

Abstract

A chemotherapy model for a malignant tumor is considered, and the optimal control (therapy) problem of minimizing the number of tumor cells at a fixed final instant is investigated. In this problem, the value function is calculated, which assigns the value (the optimal achievable result) to each initial state. An optimal feedback (optimal synthesis) is constructed, using which for any initial state ensures the achievement of the corresponding optimal result. The proposed constructions are based on the method of Cauchy characteristics, the Pontryagin maximum principle, and the theory of generalized (minimax/viscosity) solutions of the Hamilton-Jacobi-Bellman equation describing the value function.