An optimization technique for solving a class of nonlinear fractional optimal control problems: Application in cancer treatment

This paper proposes an optimization method for solving a general form of nonlinear fractional optimal control problems (NFOCP) governed by nonlinear fractional dynamical systems and mixed conditions. A model for cancer treatment based on the NFOCP is developed, by including the population of immune cells, tumor cells, normal cells, fat cells, chemotherapeutic and immunotherapeutic drug concentrations. The aim behind the NFOCP is to find the optimal doses of chemotherapeutic and immunotherapeutic drugs which minimize the difference between the numbers of tumor and normal cells. The proposed approach includes inserting weight constants for the tumor and normal cells in the cost functional so that the normal cell population is large minimizing, therefore, the tumor burden. The method approximates the solutions of the NFOCP using the generalized shifted Legendre polynomials (GSLP). Operational matrices of integer and fractional derivatives are derived for the GSLP. The Gaussian quadrature rule and the Lagrange multipliers method reduce the NFOCP to a system of algebraic equations. Additionally, the error analysis of the algorithm is considered. Numerical examples demonstrate the effectiveness of the method.