Application of the extended Chebyshev cardinal wavelets in solving fractional optimal control problems with ABC fractional derivative

Abstract

In this study, two categories of fractional optimal control problems are investigated. One category is optimal control problems, which includes a fractional dynamic system, and the other category, in addition to the fractional dynamics system, includes inequality constraints. The Atangana–Baleanu fractional derivative in the Caputo sense is used to define these problems. The extended Chebyshev cardinal wavelets as an appropriate class of basis functions are introduced to construct two numerical methods for these problems. To solve these problems, at first, an operational matrix for the Atangana–Baleanu fractional integral of these wavelets is derived. Then, by approximating the fractional derivative of the state variables and control variables in terms of the extended Chebyshev cardinal wavelets, and employing the fractional integral operational matrix of these wavelets and the Lagrange multipliers method, the problems under consideration are converted into systems of algebraic equations, which can be easily solved. To examine the accuracy of the established methods, some numerical examples are solved.

Keywords:

• Atangana–Baleanu fractional derivative
• extended Chebyshev cardinal wavelets
• fractional optimal control problems

Data availability statement

Data sharing is not applicable to this paper as no new data were created or analysed in this study.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Notes on contributors

M. H. Heydari

M. H. Heydari received his Bachelor’s Degree in Applied Mathematics from Razi University of Kermanshah in 2007, his Master’s Degree in Numerical Analysis from Yazd University in 2009, and his doctoral degree in Applied Mathematics in 2014. He is currently an Assistant Professor of Applied Mathematics in Shiraz University of Technology. His research interests are Fractional Differential Equations, Wavelets Theory, Fractional Optimal Control Problems, Stochastic Differential Equations and Meshless Methods.

R. Tavakoli

R. Tavakoli recieved his Bachelor's Degree in Urban Engineering from Shiraz University in 2015. In 2019, he commenced pursuing his favorite field of study, the science of mathematics, and eventually obtained his Master's Degree in Applied Mathematics, Numerical Analysis from Shiraz University of Technology in 2021. Currently he is pursuing his Doctorate Degree in Applied Mathematics from K. N. Toosi of Technology University in Tehran.

M. Razzaghi

M. Razzaghi received his Bachelor’s Degree in Mathematics with a minor in Physics from the University of Sussex in England, and his Master’s Degree in Applied Mathematics from the University of Waterloo in Canada. He then returned to Sussex University and earned his doctoral degree. He is currently a professor in the Department of Mathematics and Statistics in the Mississippi State University, and has served as Head of the Department since 2007.