Legendre spectral-collocation method for solving fractional optimal control of HIV infection of CD4 + T cells mathematical model

Abstract

In this paper, the optimal control problem of HIV infection is presented. We introduce fractional order into a model of HIV infection where the derivatives are taken in the sense of Caputo. The necessary optimality conditions are characterized using Pontryagin’s maximum principle. The optimality system is approximated by shifted Legendre polynomials which transform the system of differential equations into a nonlinear system algebraic equations with unknown coefficients. Newton’s iteration method is used to solve this system of nonlinear algebraic equations. The value of the objective function which is obtained by using shifted Legendre polynomials is compared with the value of the objective function which id obtained by using numerical methods such as the iterative optimal control method and forward–backward sweep method. Numerical results are also given to demonstrate the validity and applicability of the presented technique.