Novel Numerical Methods for Optimal Control Problems Involving Fractional-Order Differential Equations

This paper presents a numerical algorithm for solving a class of optimal control problems with a dynamic system containing fractional differential equations. We first propose a robust 2nd-order numerical integration scheme for the fractional system, based a set of judiciously chosen quadrature points. The objective is approximated by the trapezoidal rule. We then apply a gradient-based optimization method to solve the discretized optimal control problem. Formulas for calculating the gradients with respect to the unknown discrete control values are derived. Computational results demonstrate that the proposed method is able to generate good numerical approximations for optimal problems with multiple state and control variables. The results also show that the method is robust with respect to the fractional orders of derivatives involved in the dynamics.