The control parametrization technique for numerically solving fractal–fractional optimal control problems involving Caputo–Fabrizio derivatives

Abstract

Fractal–fractional derivatives open new opportunities for modelling complex processes. In this paper, we develop a novel numerical computation approach for solving fractal–fractional optimal control problems with Caputo–Fabrizio derivatives. Firstly, we propose a general class of fractal–fractional optimal control problems with Caputo–Fabrizio derivatives and subject to state constraints of equality and inequality. Then, by using control parametrization technique, the fractal–fractional optimal control problem is approximated by a series of finite-dimensional optimization problems. Furthermore, the gradients of the cost and constraint functions in regard to decision variables are derived, which can be obtained by solving auxiliary fractal–fractional systems. A 3rd-order numerical scheme is also presented to solve the involved fractal–fractional systems. On the basis of this result, we develop a gradient-based optimization algorithm to solve the resulting optimization problem. Finally, numerical results of solving three examples, one of which involves optimal control of acquired immunodeficiency syndrome epidemic, are provided to demonstrate the applicability and effectiveness of the developed algorithm.