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

• Published in: International journal of systems science 2022-09, Vol.53 (12), p.2694-2708
• Main Authors: Heydari, M. H., Tavakoli, R., Razzaghi, M.
• Format: Article
• Language: English
• Subjects: Atangana-Baleanu fractional derivative; Basis functions; Chebyshev approximation; Derivatives; Dynamical systems; extended Chebyshev cardinal wavelets; Fractional calculus; fractional optimal control problems; Integrals; Lagrange multiplier; Numerical methods; Optimal control
• ISSN: 0020-7721; 1464-5319
• DOI: 10.1080/00207721.2022.2058640

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.