On the Fractional Optimal Control Problems with Singular and Non–Singular Derivative Operators: A Mathematical Derive

Timothy F. Chen ,J. C. Chen ,Bunnitru Daleanu

doi:10.32861/sr.412.95.98

Abstract

The aim of this paper is to design an efficient numerical method to solve a class of time fractional optimal control problems. In this problem formulation, the fractional derivative operator is consid- ered in three cases with both singular and nonâ€“singular kernels. The necessary conditions are derived for the optimality of these problems and the proposed method is evaluated for different choices of derivative operators. Simulation results indicate that the suggested technique works well and pro- vides satisfactory results with considerably less computational time than the other existing methods. Comparative results also verify that the fractional operator with Mittagâ€“Leffler kernel in the Caputo sense improves the performance of the controlled system in terms of the transient response compared to the other fractional and integer derivative operators.

Highlights

The fractional calculus (FC), as a branch of mathematical analysis, investigates the extension of deriva- tives and integrals to non–integer orders [1,2,3]
Agrawal [9] approximated the solution of the fractional optimal control prob- lems (FOCPs) in the Riemann–Liouville sense as a truncated series
Frederico and Torres [11] applied a Noether–type theorem for the FOCPs in the Caputo sense

Summary

Introduction

The fractional calculus (FC), as a branch of mathematical analysis, investigates the extension of deriva- tives and integrals to non–integer orders [1,2,3]. Agrawal [9] approximated the solution of the fractional optimal control prob- lems (FOCPs) in the Riemann–Liouville sense as a truncated series. Almeida and Torres [12] approximated the FOCP by a new integer one and used a finite dif- ference method to solve it. Sweilam and Al-Mekhlafi [13] investigated the FOCPs via an iterative optimal control scheme together with a generalized Euler method. Ejlali and Hosseini [14] employed a new framework on the basis of the direct pseudospectral method for solving the FOCPs. Jahanshahi and Torres [15] rewrote the FOCP as a classical static optimization problem by using known formulas for the fractional derivative (FD) of polynomials, and they solved the latter problem by the Ritz method. A new iterative algorithm was examined by Jajarmi, et al [22] for the nonlinear FOCPs with external persistent disturbances

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Methods

Conclusion