Abstract
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.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Abdeljawad, T., Baleanu, D.: Integration by parts and its applications of a new nonlocal fractional derivative with Mittag-Leffler nonsingular kernel. J. Nonlinear Sci. Appl. 10, 1098–1107 (2017)
Agrawal, O.P.: Formulation of Euler-Lagrange equations for fractional variational problems. J. Math. Anal. Appl. 272, 368–379 (2002)
Agrawal, O.P.: A general formulation and solution scheme for fractional optimal control problems. Nonlinear Dyn. 38, 323–337 (2004)
Agrawal, O.P., Baleanu, D.A.: Hamiltonian formulation and direct numerical scheme for fractional optimal control problems. J. Vib. Control. 13(9–10), 1269–1281 (2007)
Agarwal, R.P., Baghli, S., Benchohra, M.: Controllability for semilinear functional and neutral functional evolution equations with infinite delay in Fréchet spaces. Appl. Math. Optim. 60, 253–274 (2009)
Atangana, A., Baleanu, D.: New fractional derivatives with non-local and non-singular kernel: theory and application to heat transfer model. Therm. Sci. 20(2), 763–769 (2016)
Bahaa, G.M.: Fractional optimal control problem for infinite order system with control constraints. Adv. Differ. Equ. 250, 1–16 (2016)
Bahaa, G.M.: Fractional optimal control problem for variational inequalities with control constraints. IMA J. Math. Control. Inf. 33(3), 1–16 (2016)
Bahaa, G.M.: Fractional optimal control problem for differential system with control constraints. Filomat 30(8), 2177–2189 (2016)
Bahaa, G.M.: Fractional optimal control problem for differential system with delay argument. Adv. Differ. Equ. 69, 1–19 (2017)
Bahaa, G.M.: Fractional optimal control problem for variable-order differential systems. Fract. Calc. Appl. Anal. 20(6), 1–16 (2017)
Bahaa, G.M., Tang, Q.: Optimal control problem for coupled time-fractional evolution systems with control constraints. J. Dyn. Differ. Equ. 1, 1–21 (2017)
Bahaa, G.M., Tang, Q.: Optimality conditions for fractional diffusion equations with weak Caputo derivatives and variational formulation. J. Fract. Calc. Appl. 9(1), 100–119 (2018)
Baleanu, D., Agrawal, O.M.P.: Fractional Hamilton formalism within Caputo’s derivative. Czechoslov. J. Phys. 56(10–11), 1087–1092 (2000)
Baleanu, D., Avkar, T.: Lagrangian with linear velocities within Riemann-Liouville fractional derivatives. Nuovo Cimento B 119, 73–79 (2004)
Baleanu, D., Muslih, S.I.: Lagrangian formulation on classical fields within Riemann-Liouville fractional derivatives. Phys. Scr. 72(2–3), 119–121 (2005)
Baleanu, D., Jajarmi, A., Hajipour, M.: A new formulation of the fractional optimal control problems involving Mittag-Leffler nonsingular kernel. J Optim. Theory Appl. 175, 718–737 (2017)
Coronel-Escamilla, A., Gómez-Aguilar, J.F., Alvarado-Méndez, E., Guerrero-Ramírez, G.V., Escobar-Jiménez, R.F.: Fractional dynamics of charged particles in magnetic fields. Int. J. Mod. Phys. C 27(08), 1–16 (2016)
Coronel-Escamilla, A., Gómez-Aguilar, J.F., Baleanu, D., Córdova-Fraga, T., Escobar-Jiménez, R.F., Olivares-Peregrino, V.H., Qurashi, M.M.A.: Bateman-Feshbach tikochinsky and Caldirola-Kanai oscillators with new fractional differentiation. Entropy 19(2), 1–21 (2017)
Cuahutenango-Barro, B., Taneco-Hernández, M.A., Gómez-Aguilar, J.F.: On the solutions of fractional-time wave equation with memory effect involving operators with regular kernel. Chaos Solitons Fractals 115, 283–299 (2018)
Djida, J.D., Atangana, A., Area, I.: Numerical computation of a fractional derivative with non-local and non-singular kernel. Math. Model. Nat. Phenom. 12(3), 4–13 (2017)
Djida, J.D., Mophou, G.M., Area, I.: Optimal control of diffusion equation with fractional time derivative with nonlocal and nonsingular Mittag-Leffler kernel (2017). arXiv:1711.09070
El-Sayed, A.M.A.: On the stochastic fractional calculus operators. J. Fract. Calc. Appl. 6(1), 101–109 (2015)
Frederico Gastao, S.F., Torres Delfim, F.M.: Fractional optimal control in the sense of Caputo and the fractional Noether’s theorem. Int. Math. Forum 3(10), 1–17 (2008)
Gómez-Aguilar, J.F.: Behavior characteristics of a cap-resistor, memcapacitor, and a memristor from the response obtained of RC and RL electrical circuits described by fractional differential equations. Turk. J. Electr. Eng. Comput. Sci. 24(3), 1–16 (2016)
Gómez-Aguilar, J.F.: Novel analytical solutions of the fractional Drude model. Optik 168, 728–740 (2018)
Gómez-Aguilar, J.F., Dumitru, B.: Fractional transmission line with losses. Zeitschrift für Naturforschung A 69(10–11), 539–546 (2014)
Gómez-Aguilar, J.F., Escobar-Jiménez, R.F., López-López, M.G., Alvarado-Martínez, V.M.: Atangana-Baleanu fractional derivative applied to electromagnetic waves in dielectric media. J. Electromagn. Waves Appl. 30(15), 1937–1952 (2016)
Gómez-Aguilar, J.F., Torres, L., Yépez-Martínez, H., Baleanu, D., Reyes, J.M., Sosa, I.O.: Fractional Liénard type model of a pipeline within the fractional derivative without singular kernel. Adv. Differ. Equ. 2016(1), 1–17 (2016)
Gómez-Aguilar, J.F., Yépez-Martínez, H., Escobar-Jiménez, R.F., Astorga-Zaragoza, C.M., Reyes-Reyes, J.: Analytical and numerical solutions of electrical circuits described by fractional derivatives. Appl. Math. Model. 40(21–22), 9079–9094 (2016)
Gómez-Aguilar, J.F., Yépez-Martínez, H., Torres-Jiménez, J., Córdova-Fraga, T., Escobar-Jiménez, R.F., Olivares-Peregrino, V.H.: Homotopy perturbation transform method for nonlinear differential equations involving to fractional operator with exponential kernel. Adv. Differ. Equ. 2017(1), 1–18 (2017)
Gómez-Aguilar, J.F., Atangana, A., Morales-Delgado, J.F.: Electrical circuits RC, LC, and RL described by Atangana-Baleanu fractional derivatives. Int. J. Circuit Theory Appl. 1, 1–22 (2017)
Hafez, F.M., El-Sayed, A.M.A., El-Tawil, M.A.: On a stochastic fractional calculus. Fract. Calc. Appl. Anal. 4(1), 81–90 (2001)
Jarad, F., Maraba, T., Baleanu, D.: Fractional variational optimal control problems with delayed arguments. Nonlinear Dyn. 62, 609–614 (2010)
Jarad, F., Maraba, T., Baleanu, D.: Higher order fractional variational optimal control problems with delayed arguments. Appl. Math. Comput. 218, 9234–9240 (2012)
Kilbas, A.A., Saigo, M., Saxena, K.: Generalized Mittag-Leffler function and generalized fractional calculus operators. Integr. Transform. Spec. Funct. 15(1), 1–13 (2004)
Mophou, G.M.: Optimal control of fractional diffusion equation with state constraints. Comput. Math. Appl. 62, 1413–1426 (2011)
Morales-Delgado, V.F., Taneco-Hernández, M.A., Gómez-Aguilar, J.F.: On the solutions of fractional order of evolution equations. Eur. Phys. J. Plus 132(1), 1–17 (2017)
Ozdemir, N., Karadeniz, D., Iskender, B.B.: Fractional optimal control problem of a distributed system in cylindrical coordinates. Phys. Lett. A 373, 221–226 (2009)
Riewe, F.: Nonconservative Lagrangian and Hamiltonian mechanics. Phys. Rev. E 53, 1890–1899 (1996)
Riewe, F.: Mechanics with fractional derivatives. Phys. Rev. E 55, 3581–3592 (1997)
Saad, K.M., Gómez-Aguilar, J.F.: Analysis of reaction diffusion system via a new fractional derivative with non-singular kernel. Phys. A Stat. Mech. Appl. 509, 703–716 (2018)
Yépez-Martínez, H., Gómez-Aguilar, J.F., Sosa, I.O., Reyes, J.M., Torres-Jiménez, J.: The Feng’s first integral method applied to the nonlinear mKdV space-time fractional partial differential equation. Rev. Mex. Fís 62(4), 310–316 (2016)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2019 Springer Nature Switzerland AG
About this chapter
Cite this chapter
Bahaa, G.M., Atangana, A. (2019). Necessary and Sufficient Optimality Conditions for Fractional Problems Involving Atangana–Baleanu’s Derivatives. In: Gómez, J., Torres, L., Escobar, R. (eds) Fractional Derivatives with Mittag-Leffler Kernel. Studies in Systems, Decision and Control, vol 194. Springer, Cham. https://doi.org/10.1007/978-3-030-11662-0_2
Download citation
DOI: https://doi.org/10.1007/978-3-030-11662-0_2
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-11661-3
Online ISBN: 978-3-030-11662-0
eBook Packages: EngineeringEngineering (R0)