Direct transcription methods based on fractional integral approximation formulas for solving nonlinear fractional optimal control problems

TL;DR

This paper introduces three direct numerical methods using fractional integral approximation formulas to efficiently solve nonlinear fractional optimal control problems without deriving necessary conditions.

Contribution

The paper develops simple, quick, and reliable direct methods based on Grünwald-Letnikov, trapezoidal, and Simpson formulas for solving FOCPs, avoiding complex derivations.

Findings

Methods successfully solve various FOCPs including free final time and path constraints.

Numerical tests demonstrate high efficiency and reliability of the proposed approaches.

Closed-form gradients and Jacobians enhance computational performance.

Abstract

This paper presents three direct methods based on Gr\"{u}nwald-Letnikov, trapezoidal and Simpson fractional integral formulas to solve fractional optimal control problems (FOCPs). At first, the fractional integral form of FOCP is considered, then the fractional integral is approximated by Gr\"{u}nwald-Letnikov, trapezoidal and Simpson formulas in a matrix approach. Thereafter, the performance index is approximated either by trapezoidal or Simpson quadrature. As a result, FOCP are reduced to nonlinear programming problems, which can be solved by many well-developed algorithms. To improve the efficiency of the presented method, the gradient of the objective function and the Jacobian of constraints are prepared in closed forms. It is pointed out that the implementation of the methods is simple and, due to the fact that there is no need to derive necessary conditions, the methods can be simply and quickly used to solve a wide class of FOCPs. The efficiency and reliability of the presented methods are assessed by ample numerical tests involving a free final time with path constraint FOCP, a bang-bang FOCP and an optimal control of a fractional-order HIV-immune system.