{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,14]],"date-time":"2026-04-14T01:26:39Z","timestamp":1776129999795,"version":"3.50.1"},"reference-count":63,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2025,3,10]],"date-time":"2025-03-10T00:00:00Z","timestamp":1741564800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>The objective of this work is to discuss and thoroughly analyze the fractional variational principles of symmetric systems involving distributed-order Atangana\u2013Baleanu derivatives. A component of distributed order, the fractional Euler\u2013Lagrange equations of fractional Lagrangians for constrained systems are studied concerning Atangana\u2013Baleanu derivatives. We give a general formulation and a solution technique for a class of fractional optimal control problems (FOCPs) for such systems. The dynamic constraints are defined by a collection of FDEs, and the performance index of an FOCP is considered a function of the control variables and the state. The formula for fractional integration by parts, the Lagrange multiplier, and the calculus of variations are used to obtain the Euler\u2013Lagrange equations for the FOCPs.<\/jats:p>","DOI":"10.3390\/sym17030417","type":"journal-article","created":{"date-parts":[[2025,3,10]],"date-time":"2025-03-10T12:02:59Z","timestamp":1741608179000},"page":"417","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":7,"title":["Fractional Optimal Control Problem for Symmetric System Involving Distributed-Order Atangana\u2013Baleanu Derivatives with Non-Singular Kernel"],"prefix":"10.3390","volume":"17","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-6670-3110","authenticated-orcid":false,"given":"Bahaa Gaber","family":"Mohamed","sequence":"first","affiliation":[{"name":"Department of Mathematics and Computer Science, Faculty of Sciences, Beni-Suef University, Beni-Suef 62511, Egypt"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5127-7822","authenticated-orcid":false,"given":"Ahlam Hasan","family":"Qamlo","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Sciences, Umm Al-Qura University, Makkah 21955, Saudi Arabia"}]}],"member":"1968","published-online":{"date-parts":[[2025,3,10]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"763","DOI":"10.2298\/TSCI160111018A","article-title":"New fractional derivatives with non-local and non-singular kernel: Theory and application to heat transfer model","volume":"20","author":"Atangana","year":"2016","journal-title":"Therm. Sci."},{"key":"ref_2","first-page":"73","article-title":"A new definition of fractional derivative without singular kernel","volume":"1","author":"Caputo","year":"2001","journal-title":"Prog. Fract. Differ. Appl."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"247","DOI":"10.1186\/s13662-017-1306-z","article-title":"On a new class of fractional operators","volume":"2017","author":"Jarad","year":"2017","journal-title":"Adv. Differ. Equ."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"129","DOI":"10.1016\/j.chaos.2019.03.001","article-title":"Optimal control problem for variable-order fractional differential systems with time delay involving Atangana-Baleanu derivatives","volume":"122","author":"Bahaa","year":"2019","journal-title":"Chaos Solitons Fractals (Nonlinear Sci. Nonequilib. Complex Phenom.)"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"102244","DOI":"10.1016\/j.jocs.2024.102244","article-title":"The Galerkin Bell method to solve the fractional optimal control problems with inequality constraints","volume":"77","author":"Sadek","year":"2024","journal-title":"J. Comput. Sci."},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Sadek, L. (2024). Control theory for fractional differential Sylvester matrix equations with Caputo fractional derivative. J. Vib. Control.","DOI":"10.1177\/10775463241246430"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"103352","DOI":"10.1016\/j.jksus.2024.103352","article-title":"Introducing novel \u03b8-fractional operators: Advances in fractional calculus","volume":"36","author":"Sadek","year":"2024","journal-title":"J. King Saud Univ.-Sci."},{"key":"ref_8","doi-asserted-by":"crossref","unstructured":"Sadek, L. (2023). A cotangent fractional derivative with the application. Fractal Fract., 7.","DOI":"10.3390\/fractalfract7060444"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"1890","DOI":"10.1103\/PhysRevE.53.1890","article-title":"Nonconservative Lagrangian and Hamiltonian mechanics","volume":"53","author":"Riewe","year":"1996","journal-title":"Phys. Rev. E"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"3581","DOI":"10.1103\/PhysRevE.55.3581","article-title":"Mechanics with fractional derivatives","volume":"55","author":"Riewe","year":"1997","journal-title":"Phys. Rev. E"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"368","DOI":"10.1016\/S0022-247X(02)00180-4","article-title":"Formulation of Euler-Lagrange equations for fractional variational problems","volume":"272","author":"Agrawal","year":"2002","journal-title":"J. Math. Anal. Appl."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"323","DOI":"10.1007\/s11071-004-3764-6","article-title":"A general formulation and solution scheme for fractional optimal control problems","volume":"38","author":"Agrawal","year":"2004","journal-title":"Nonlinear Dyn."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"1269","DOI":"10.1177\/1077546307077467","article-title":"Hamiltonian formulation and direct numerical scheme for fractional optimal control problems","volume":"13","author":"Agrawal","year":"2007","journal-title":"J. Vib. Control"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"119","DOI":"10.1238\/Physica.Regular.072a00119","article-title":"Lagrangian formulation on classical fields within Riemann-Liouville fractional derivatives","volume":"72","author":"Baleanu","year":"2005","journal-title":"Phys. Scr."},{"key":"ref_15","first-page":"73","article-title":"Lagrangian with linear velocities within Riemann-Liouville fractional derivatives","volume":"119","author":"Baleanu","year":"2004","journal-title":"Nuovo Cimnto B"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"1087","DOI":"10.1007\/s10582-006-0406-x","article-title":"Fractional Hamilton formalism within Caputo\u2019s derivative","volume":"56","author":"Baleanu","year":"2000","journal-title":"Czechoslov. J. Phys."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"718","DOI":"10.1007\/s10957-017-1186-0","article-title":"A new formulation of the fractional optimal control problems involving Mittag-Leffler nonsingular kernel","volume":"175","author":"Baleanu","year":"2017","journal-title":"J. Optim. Theory. Appl."},{"key":"ref_18","doi-asserted-by":"crossref","unstructured":"Area, I., Cabada, A., Cid, J.\u00c1., Franco, D., Liz, E., Pouso, R.L., and Rodr\u00edguez-L\u00f3pez, R. (2019). Time-Fractional Optimal Control of Initial Value Problems on Time Scales. Nonlinear Analysis and Boundary Value Problems, Springer International Publishing AG.","DOI":"10.1007\/978-3-030-26987-6"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"68","DOI":"10.1016\/j.camwa.2010.10.030","article-title":"Optimal control of fractional diffusion equation","volume":"61","author":"Mophou","year":"2011","journal-title":"Comput. Math. Appl."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"1413","DOI":"10.1016\/j.camwa.2011.04.044","article-title":"Optimal control of fractional diffusion equation with state constraints","volume":"62","author":"Mophou","year":"2011","journal-title":"Comput. Math. Appl."},{"key":"ref_21","doi-asserted-by":"crossref","unstructured":"Abdel-Gaid, S.H., Qamlo, A.H., and Bahaa, G.M. (2024). Bang-bang property and time optimal control for Caputo fractional differential systems. Fractal Fract., 8.","DOI":"10.3390\/fractalfract8020084"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"1447","DOI":"10.1515\/fca-2017-0076","article-title":"Fractional optimal control problem for variable-order differential systems","volume":"20","author":"Bahaa","year":"2017","journal-title":"Fract. Calc. Appl. Anal."},{"key":"ref_23","doi-asserted-by":"crossref","unstructured":"G\u00f3mez, J.F., Torres, L., and Escobar, R.F. (2019). Necessary and Sufficient Optimality Conditions for Fractional Problems Involving Atangana-Baleanu\u2019s Derivatives. Fractional Derivatives with Mittag-Leffler Kernel, Springer Nature Switzerland AG. Studies in Systems, Decision and Control.","DOI":"10.1007\/978-3-030-11662-0"},{"key":"ref_24","doi-asserted-by":"crossref","unstructured":"Daftardar-Gejji, V. (2019). On Mittag-Leffler Kernel-Dependent Fractional Operators with Variable Order. Fractional Calculus and Fractional Differential Equations, Springer Nature Singapore Pte Ltd.","DOI":"10.1007\/978-981-13-9227-6"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"408","DOI":"10.1016\/j.matcom.2025.01.009","article-title":"Fractional truncated exponential method for linear fractional optimal control problems","volume":"232","author":"Ounamane","year":"2025","journal-title":"Math. Comput. Simul."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"602","DOI":"10.1002\/rnc.6990","article-title":"Robust fractional order singular Kalman filter","volume":"34","author":"Nosrati","year":"2023","journal-title":"Int. J. Robust Nonlinear Control"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"1043","DOI":"10.1515\/fca-2017-0054","article-title":"Analytic approximate solutions for a class of variable order fractional differential equations using the polynomial least squares method","volume":"20","author":"Bota","year":"2017","journal-title":"Fract. Calc. Appl. Anal."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"73","DOI":"10.1007\/BF02826009","article-title":"Mean fractional-order-derivatives differential equations and filters","volume":"41","author":"Caputo","year":"1995","journal-title":"Ann. dell\u2019Univ. Ferrara"},{"key":"ref_29","first-page":"421","article-title":"Distributed order differential equations modelling dielectric induction and diffusion","volume":"4","author":"Caputo","year":"2001","journal-title":"Fract. Calc. Appl. Anal."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"046129","DOI":"10.1103\/PhysRevE.66.046129","article-title":"Retarding subdiffusion and accelerating superdiffusion governed by distributed-order fractional diffusion equations","volume":"66","author":"Chechkin","year":"2002","journal-title":"Phys. Rev. E"},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"4","DOI":"10.1051\/mmnp\/201712302","article-title":"Numerical Computation of a Fractional Derivative with Non-Local and Non-Singular Kernel","volume":"12","author":"Djida","year":"2017","journal-title":"Math. Model. Nat. Phenom."},{"key":"ref_32","doi-asserted-by":"crossref","unstructured":"Djida, J.D., Mophou, G.M., and Area, I. (2017). Optimal control of diffusion equation with fractional time derivative with nonlocal and nonsingular mittag-leffler kernel. arXiv.","DOI":"10.1007\/s10957-018-1305-6"},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"031136","DOI":"10.1103\/PhysRevE.83.031136","article-title":"Fractional Langevin equations of distributed order","volume":"83","author":"Eab","year":"2011","journal-title":"Phys. Rev. E"},{"key":"ref_34","first-page":"479","article-title":"Fractional optimal control in the sense of Caputo and the fractional Noether\u2019s theorem","volume":"3","author":"Frederico","year":"2008","journal-title":"Int. Math. Forum"},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"2973","DOI":"10.1016\/j.camwa.2012.01.053","article-title":"Distributed order equations as boundary value problems","volume":"64","author":"Ford","year":"2012","journal-title":"Comput. Math. Appl."},{"key":"ref_36","first-page":"179","article-title":"Irving-Mullineux oscillator via fractional derivatives with Mittag-Leffler kernel","volume":"35","year":"2017","journal-title":"Chaos Solitons Fractals"},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"562","DOI":"10.1016\/j.physa.2016.08.072","article-title":"Space-time fractional diffusion equation using a derivative with nonsingular and regular kernel","volume":"465","year":"2017","journal-title":"Phys. A Stat. Mech. Its Appl."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"1514","DOI":"10.1002\/cta.2348","article-title":"Electrical circuits RC, LC, and RL described by Atangana-Baleanu fractional derivatives","volume":"45","author":"Atangana","year":"2017","journal-title":"Int. J. Circ. Theor. Appl."},{"key":"ref_39","unstructured":"Hartley, T.T., and Lorenzo, C.F. (1999). Fractional System Identification: An Approach Using Continuous Order-Distributions, Technical Report."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"4624","DOI":"10.1002\/asjc.3127","article-title":"Fractional optimal control problems with both integer-order and Atangana\u2013Baleanu Caputo derivatives","volume":"25","author":"Ma","year":"2023","journal-title":"Asian J. Control"},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"334","DOI":"10.1186\/s13662-020-02793-9","article-title":"Optimal control for cancer treatment mathematical model using Atangana\u2013Baleanu\u2013Caputo fractional derivative","volume":"2020","author":"Sweilam","year":"2020","journal-title":"Adv. Differ. Equations"},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"96","DOI":"10.1002\/oca.2664","article-title":"Optimal control problems with Atangana-Baleanu fractional derivative","volume":"42","author":"Tajadodi","year":"2021","journal-title":"Optim. Control Appl. Meth."},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"57","DOI":"10.1023\/A:1016586905654","article-title":"Variable order and distributed order fractional operators","volume":"29","author":"Lorenzo","year":"2002","journal-title":"Nonlin. Dyn."},{"key":"ref_44","first-page":"1869","article-title":"Time distributed-order diffusion-wave equation. I. Volterra-type equation","volume":"465","author":"Atanackovic","year":"2009","journal-title":"Proc. R. Soc. A Math. Phys. Eng. Sci."},{"key":"ref_45","first-page":"865","article-title":"On the existence of the order domain and the solution of distributed order equations\u2014Part I","volume":"2","author":"Bagley","year":"2000","journal-title":"Int. J. Appl. Math."},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"609","DOI":"10.1007\/s11071-010-9748-9","article-title":"Fractional variational optimal control problems with delayed arguments","volume":"62","author":"Jarad","year":"2010","journal-title":"Nonlinear Dyn."},{"key":"ref_47","first-page":"9234","article-title":"Higher order fractional variational optimal control problems with delayed arguments","volume":"218","author":"Jarad","year":"2012","journal-title":"Appl. Math. Comput."},{"key":"ref_48","doi-asserted-by":"crossref","first-page":"2596","DOI":"10.1177\/10775463211016967","article-title":"A numerical approach for solving fractional optimal control problems with Mittag-Leffler kernel","volume":"28","author":"Jafari","year":"2021","journal-title":"J. Vibr. Control"},{"key":"ref_49","doi-asserted-by":"crossref","unstructured":"Jiao, Z., Chen, Y., and Podlubny, I. (2012). Distributed-Order Dynamic Systems: Stability, Simulation, Applications and Perspectives, Springer.","DOI":"10.1007\/978-1-4471-2852-6"},{"key":"ref_50","doi-asserted-by":"crossref","first-page":"188","DOI":"10.1515\/fca-2016-0048","article-title":"Maximum principles for multi-term space-time variable-order fractional diffusion equations and their applications","volume":"19","author":"Liu","year":"2016","journal-title":"Fract. Calc. Appl. Anal."},{"key":"ref_51","doi-asserted-by":"crossref","first-page":"1267","DOI":"10.1177\/1077546307087452","article-title":"Time fractional diffusion of distributed order","volume":"14","author":"Mainardi","year":"2008","journal-title":"J. Vib. Control"},{"key":"ref_52","doi-asserted-by":"crossref","first-page":"1215","DOI":"10.1016\/j.spa.2006.01.006","article-title":"Stochastic model for ultraslow diffusion","volume":"116","author":"Meerschaert","year":"2006","journal-title":"Stoch. Process. Appl."},{"key":"ref_53","first-page":"127950","article-title":"Extended fractional singular Kalman filter","volume":"448","author":"Nosrati","year":"2023","journal-title":"Appl. Math. Comput."},{"key":"ref_54","doi-asserted-by":"crossref","first-page":"221","DOI":"10.1016\/j.physleta.2008.11.019","article-title":"Fractional optimal control problem of a distributed system in cylindrical coordinates","volume":"373","author":"Ozdemir","year":"2009","journal-title":"Phys. Lett. A"},{"key":"ref_55","doi-asserted-by":"crossref","first-page":"042117","DOI":"10.1103\/PhysRevE.92.042117","article-title":"Distributed-order diffusion equations and multifractality: Models and solutions","volume":"92","author":"Sandev","year":"2015","journal-title":"Phys. Rev. E"},{"key":"ref_56","first-page":"1323","article-title":"Distributed-order fractional kinetics","volume":"35","author":"Sokolov","year":"2004","journal-title":"Acta Phys. Pol. B"},{"key":"ref_57","doi-asserted-by":"crossref","first-page":"3525","DOI":"10.1007\/s40314-017-0530-1","article-title":"A Legendre spectral quadrature tau method for the multi-term time-fractional diffusion equations","volume":"37","author":"Zaky","year":"2017","journal-title":"Comp. Appl. Math."},{"key":"ref_58","doi-asserted-by":"crossref","first-page":"77","DOI":"10.1007\/BF01171449","article-title":"A generalized model for the uniaxial isothermal deformation of a viscoelastic body","volume":"159","author":"Atanackovic","year":"2002","journal-title":"Acta Mech."},{"key":"ref_59","first-page":"409","article-title":"Boundary value problems for the generalized time-fractional diffusion equation of distributed order","volume":"12","author":"Luchko","year":"2009","journal-title":"Fract. Calc. Appl. Anal."},{"key":"ref_60","doi-asserted-by":"crossref","first-page":"1098","DOI":"10.22436\/jnsa.010.03.20","article-title":"Integration by parts and its applications of a new nonlocal fractional derivative with Mittag-Leffler nonsingular kernel","volume":"10","author":"Abdeljawad","year":"2017","journal-title":"J. Nonlinear Sci. Appl."},{"key":"ref_61","doi-asserted-by":"crossref","first-page":"31","DOI":"10.1080\/10652460310001600717","article-title":"Generalized Mittag-Leffler function and generalized fractional calculus operators","volume":"15","author":"Kilbas","year":"2004","journal-title":"Int. Tran. Spec. Funct."},{"key":"ref_62","doi-asserted-by":"crossref","first-page":"252","DOI":"10.1016\/j.jmaa.2007.08.024","article-title":"Distributed order calculus and equations of ultraslow diffusion","volume":"340","author":"Kochubei","year":"2008","journal-title":"J. Math. Anal. Appl."},{"key":"ref_63","doi-asserted-by":"crossref","first-page":"253","DOI":"10.1007\/s00245-009-9073-1","article-title":"Controllability for semilinear functional and neutral functional evolution equations with infinite delay in Frechet spaces","volume":"60","author":"Agarwal","year":"2009","journal-title":"Appl. Math. 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