optimal control for differential equations

In this article different numerical techniques for solving optimal control problems is introduced, the aim of this paper is to achieve the best accuracy for the Optimal Control Problem (OCP) which has the objective of minimizing the size of tumor cells by the end of the treatment. An important aspect is considered, which is, the optimal concentrations of drugs that not affect the patient’s health significantly. To study the behavior of tumor growth, a mathematical model is used to simulate the dynamic behavior of tumors since it is difficult to prototype dynamic behavior of the tumor. A tumor-immune model with four components, namely, tumor cells, active cytotoxic T-cells (CTLs), helper T-cells, and a chemotherapeutic drug is used. Two general categories of optimal control methods which are indirect methods and direct ones based on nonlinear programming solvers and interior point algorithms are compared. Within the direct optimal control techniques, we review three different solutions techniques namely (i) multiple shooting methods, (ii) trapezoidal direct collocation method, (iii) Hermit- Simpson’s collocation method and within the indirect methods we review the Pontryagin’s Maximum principle with both collocation method and the backward forward sweep method. Results show that the direct methods achieved better control than indirect methods.