A Central Difference Numerical Scheme for Fractional Optimal Control Problems

TL;DR

This paper introduces a central difference numerical scheme for fractional optimal control problems using a modified Grünwald-Letnikov approach, demonstrating convergence and stability in solving fractional differential equations.

Contribution

It proposes a novel central difference scheme for FOCPs based on a modified Grünwald-Letnikov method, improving stability and convergence in numerical solutions.

Findings

Solutions converge as sub-domain sizes decrease.

Method recovers integer-order solutions when derivative order approaches an integer.

Numerical examples validate the scheme's effectiveness.

Abstract

This paper presents a modified numerical scheme for a class of Fractional Optimal Control Problems (FOCPs) formulated in Agrawal (2004) where a Fractional Derivative (FD) is defined in the Riemann-Liouville sense. In this scheme, the entire time domain is divided into several sub-domains, and a fractional derivative (FDs) at a time node point is approximated using a modified Gr\"{u}nwald-Letnikov approach. For the first order derivative, the proposed modified Gr\"{u}nwald-Letnikov definition leads to a central difference scheme. When the approximations are substituted into the Fractional Optimal Control (FCO) equations, it leads to a set of algebraic equations which are solved using a direct numerical technique. Two examples, one time-invariant and the other time-variant, are considered to study the performance of the numerical scheme. Results show that 1) as the order of the derivative approaches an integer value, these formulations lead to solutions for integer order system, and 2) as the sizes of the sub-domains are reduced, the solutions converge. It is hoped that the present scheme would lead to stable numerical methods for fractional differential equations and optimal control problems.