Received: 13 April 2017 Revised: 22 September 2017 Accepted: 2 November 2017
An approximate method for solving fractional optimal
control problems by the hybrid of block-pulse functions and
Department of Mathematics and
Statistics, Faculty of Science, Prince of
Songkla University, Hat Yai, Thailand
Department of Mathematics and
Statistics, Mississippi State University,
Mississippi State, MS, USA
M. Razzaghi, Department of Mathematics
and Statistics, Mississippi State University,
Mississippi State, MS 39762, USA.
Faculty of Science Research Fund, Faculty
of Science, Prince of Songkla University
In this paper, an efficient and accurate computational method based on the
hybrid of block-pulse functions and Taylor polynomials is proposed for solving
a class of fractional optimal control problems. In the proposed method, the
Riemann-Liouville fractional integral operator for the hybrid of block-pulse
functions and Taylor polynomials is given. By taking into account the property
of this operator, the solution of fractional optimal control problems under
consideration is reduced to a nonlinear programming one to which existing
well-developed algorithms may be applied. The present method applies to both
fractional optimal control problems with or without inequality constraints. The
method is computationally very attractive and gives very accurate results. Easy
implementation and simple operations are the essential features of the proposed
hybrid functions. Illustrative examples are given to assess the effectiveness of
the developed approximation technique.
block pulse, Caputo derivative, fractional optimal control, hybrid functions, Taylor basis
Although the concept of fractional derivatives was introduced in the middle of the 19th century by Riemann and Liou-
ville, the book by Oldham and Spanier
, published in 1974, was the first work devoted exclusively to the subject of
fractional calculus. A history of the development of fractional calculus was given in the works of Miller and Ross
Machado et al.
In recent years, fractional calculus has drawn increasing attention and interest due to its important applications in
various fields of science and engineering (see, for example, other works
and the references therein).
Optimal control problems represent a set of differential equations describing the paths of the control variables that
minimize a function of the state and control variables.
The optimal control problems in which the criterion or the
differential equations of the dynamic systems display at least one fractional derivative lead to fractional optimal control
In recent years, FOCPs have gained importance due to their many applications in various fields.
To mention a few, FOCPs have been used in the analog fractional-order controller in temperature and motor control
a fractional adaptation scheme for lateral control of an autonomous guided vehicle,
fractional control of
heat diffusion systems,
and a fractional-order HIV-immune system with memory.
Although the optimal control theory is an area in mathematics that has been under development for many years, the
fractional optimal control theory is a very new area in mathematics.
Optim Control Appl Meth. 2018;39:873–887. wileyonlinelibrary.com/journal/oca Copyright © 2017 John Wiley & Sons, Ltd. 873