Advertisement

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

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

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.

Keywords

Fractional calculus Atangana–Baleanu fractional derivative Fractional optimal control problems 

References

  1. 1.
    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)MathSciNetGoogle Scholar
  2. 2.
    Agrawal, O.P.: Formulation of Euler-Lagrange equations for fractional variational problems. J. Math. Anal. Appl. 272, 368–379 (2002)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Agrawal, O.P.: A general formulation and solution scheme for fractional optimal control problems. Nonlinear Dyn. 38, 323–337 (2004)MathSciNetzbMATHGoogle Scholar
  4. 4.
    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)MathSciNetzbMATHGoogle Scholar
  5. 5.
    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)MathSciNetzbMATHGoogle Scholar
  6. 6.
    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)Google Scholar
  7. 7.
    Bahaa, G.M.: Fractional optimal control problem for infinite order system with control constraints. Adv. Differ. Equ. 250, 1–16 (2016)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Bahaa, G.M.: Fractional optimal control problem for variational inequalities with control constraints. IMA J. Math. Control. Inf. 33(3), 1–16 (2016)MathSciNetGoogle Scholar
  9. 9.
    Bahaa, G.M.: Fractional optimal control problem for differential system with control constraints. Filomat 30(8), 2177–2189 (2016)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Bahaa, G.M.: Fractional optimal control problem for differential system with delay argument. Adv. Differ. Equ. 69, 1–19 (2017)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Bahaa, G.M.: Fractional optimal control problem for variable-order differential systems. Fract. Calc. Appl. Anal. 20(6), 1–16 (2017)MathSciNetzbMATHGoogle Scholar
  12. 12.
    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)Google Scholar
  13. 13.
    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)MathSciNetGoogle Scholar
  14. 14.
    Baleanu, D., Agrawal, O.M.P.: Fractional Hamilton formalism within Caputo’s derivative. Czechoslov. J. Phys. 56(10–11), 1087–1092 (2000)MathSciNetGoogle Scholar
  15. 15.
    Baleanu, D., Avkar, T.: Lagrangian with linear velocities within Riemann-Liouville fractional derivatives. Nuovo Cimento B 119, 73–79 (2004)Google Scholar
  16. 16.
    Baleanu, D., Muslih, S.I.: Lagrangian formulation on classical fields within Riemann-Liouville fractional derivatives. Phys. Scr. 72(2–3), 119–121 (2005)MathSciNetzbMATHGoogle Scholar
  17. 17.
    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)MathSciNetzbMATHGoogle Scholar
  18. 18.
    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)MathSciNetGoogle Scholar
  19. 19.
    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)Google Scholar
  20. 20.
    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)MathSciNetGoogle Scholar
  21. 21.
    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)MathSciNetzbMATHGoogle Scholar
  22. 22.
    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
  23. 23.
    El-Sayed, A.M.A.: On the stochastic fractional calculus operators. J. Fract. Calc. Appl. 6(1), 101–109 (2015)MathSciNetGoogle Scholar
  24. 24.
    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)MathSciNetzbMATHGoogle Scholar
  25. 25.
    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)Google Scholar
  26. 26.
    Gómez-Aguilar, J.F.: Novel analytical solutions of the fractional Drude model. Optik 168, 728–740 (2018)Google Scholar
  27. 27.
    Gómez-Aguilar, J.F., Dumitru, B.: Fractional transmission line with losses. Zeitschrift für Naturforschung A 69(10–11), 539–546 (2014)Google Scholar
  28. 28.
    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)Google Scholar
  29. 29.
    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)zbMATHGoogle Scholar
  30. 30.
    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)MathSciNetGoogle Scholar
  31. 31.
    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)MathSciNetzbMATHGoogle Scholar
  32. 32.
    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)Google Scholar
  33. 33.
    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)MathSciNetzbMATHGoogle Scholar
  34. 34.
    Jarad, F., Maraba, T., Baleanu, D.: Fractional variational optimal control problems with delayed arguments. Nonlinear Dyn. 62, 609–614 (2010)MathSciNetzbMATHGoogle Scholar
  35. 35.
    Jarad, F., Maraba, T., Baleanu, D.: Higher order fractional variational optimal control problems with delayed arguments. Appl. Math. Comput. 218, 9234–9240 (2012)MathSciNetzbMATHGoogle Scholar
  36. 36.
    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)MathSciNetzbMATHGoogle Scholar
  37. 37.
    Mophou, G.M.: Optimal control of fractional diffusion equation with state constraints. Comput. Math. Appl. 62, 1413–1426 (2011)MathSciNetzbMATHGoogle Scholar
  38. 38.
    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)Google Scholar
  39. 39.
    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)MathSciNetzbMATHGoogle Scholar
  40. 40.
    Riewe, F.: Nonconservative Lagrangian and Hamiltonian mechanics. Phys. Rev. E 53, 1890–1899 (1996)MathSciNetGoogle Scholar
  41. 41.
    Riewe, F.: Mechanics with fractional derivatives. Phys. Rev. E 55, 3581–3592 (1997)MathSciNetGoogle Scholar
  42. 42.
    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)MathSciNetGoogle Scholar
  43. 43.
    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)MathSciNetGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

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

Personalised recommendations