Necessary and Sufficient Optimality Conditions for Fractional Problems Involving Atangana–Baleanu’s Derivatives

  • G. M. BahaaEmail author
  • A. Atangana
Part of the Studies in Systems, Decision and Control book series (SSDC, volume 194)


Recently, Atangana and Baleanu proposed a derivative with fractional order to answer some outstanding questions that were posed by many researchers within the field of fractional calculus. Their derivative has a non-singular and nonlocal kernel. In this chapter, the necessary and sufficient optimality conditions for systems involving Atangana–Baleanu’s derivatives are discussed. The fractional Euler–Lagrange equations of fractional Lagrangians for constrained systems that contains a fractional Atangana–Baleanu’s derivatives are investigated. The fractional contains both the fractional derivatives and the fractional integrals in the sense of Atangana–Baleanu. We present a general formulation and a solution scheme for a class of Fractional Optimal Control Problems (FOCPs) for those systems. The calculus of variations, the Lagrange multiplier, and the formula for fractional integration by parts are used to obtain Euler–Lagrange equations for the FOCP.


Fractional calculus Atangana–Baleanu fractional derivative Fractional optimal control problems 


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Authors and Affiliations

  1. 1.Department of Mathematics and Computer Science, Faculty of ScienceBeni-Suef UniversityBeni-SuefEgypt
  2. 2.Department of Mathematics, Faculty of ScienceTaibah UniversityAl-Madinah Al-MunawarahSaudi Arabia
  3. 3.Faculty of Natural and Agricultural SciencesInstitute of Groundwater Studies, University of Free StateBloemfonteinSouth Africa

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