Skip to main content
SpringerLink
Log in

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

  • Chapter
  • First Online:
Fractional Derivatives with Mittag-Leffler Kernel

Part of the book series: Studies in Systems, Decision and Control ((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.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  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)

    Article  MathSciNet  MATH  Google Scholar 

  2. Agrawal, O.P.: Formulation of Euler-Lagrange equations for fractional variational problems. J. Math. Anal. Appl. 272, 368–379 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  3. Agrawal, O.P.: A general formulation and solution scheme for fractional optimal control problems. Nonlinear Dyn. 38, 323–337 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  Google Scholar 

  7. Bahaa, G.M.: Fractional optimal control problem for infinite order system with control constraints. Adv. Differ. Equ. 250, 1–16 (2016)

    MathSciNet  MATH  Google Scholar 

  8. Bahaa, G.M.: Fractional optimal control problem for variational inequalities with control constraints. IMA J. Math. Control. Inf. 33(3), 1–16 (2016)

    MathSciNet  Google Scholar 

  9. Bahaa, G.M.: Fractional optimal control problem for differential system with control constraints. Filomat 30(8), 2177–2189 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  10. Bahaa, G.M.: Fractional optimal control problem for differential system with delay argument. Adv. Differ. Equ. 69, 1–19 (2017)

    MathSciNet  MATH  Google Scholar 

  11. Bahaa, G.M.: Fractional optimal control problem for variable-order differential systems. Fract. Calc. Appl. Anal. 20(6), 1–16 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  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. 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)

    MathSciNet  Google Scholar 

  14. Baleanu, D., Agrawal, O.M.P.: Fractional Hamilton formalism within Caputo’s derivative. Czechoslov. J. Phys. 56(10–11), 1087–1092 (2000)

    MathSciNet  Google Scholar 

  15. Baleanu, D., Avkar, T.: Lagrangian with linear velocities within Riemann-Liouville fractional derivatives. Nuovo Cimento B 119, 73–79 (2004)

    Google Scholar 

  16. Baleanu, D., Muslih, S.I.: Lagrangian formulation on classical fields within Riemann-Liouville fractional derivatives. Phys. Scr. 72(2–3), 119–121 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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. El-Sayed, A.M.A.: On the stochastic fractional calculus operators. J. Fract. Calc. Appl. 6(1), 101–109 (2015)

    MathSciNet  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  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. Gómez-Aguilar, J.F.: Novel analytical solutions of the fractional Drude model. Optik 168, 728–740 (2018)

    Article  Google Scholar 

  27. Gómez-Aguilar, J.F., Dumitru, B.: Fractional transmission line with losses. Zeitschrift für Naturforschung A 69(10–11), 539–546 (2014)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  MATH  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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. 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)

    MathSciNet  MATH  Google Scholar 

  34. Jarad, F., Maraba, T., Baleanu, D.: Fractional variational optimal control problems with delayed arguments. Nonlinear Dyn. 62, 609–614 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  35. Jarad, F., Maraba, T., Baleanu, D.: Higher order fractional variational optimal control problems with delayed arguments. Appl. Math. Comput. 218, 9234–9240 (2012)

    MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  37. Mophou, G.M.: Optimal control of fractional diffusion equation with state constraints. Comput. Math. Appl. 62, 1413–1426 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  40. Riewe, F.: Nonconservative Lagrangian and Hamiltonian mechanics. Phys. Rev. E 53, 1890–1899 (1996)

    Article  MathSciNet  Google Scholar 

  41. Riewe, F.: Mechanics with fractional derivatives. Phys. Rev. E 55, 3581–3592 (1997)

    Article  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  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)

    MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics and Computer Science, Faculty of Science, Beni-Suef University, Beni-Suef, Egypt

    G. M. Bahaa

  2. Department of Mathematics, Faculty of Science, Taibah University, Al-Madinah Al-Munawarah, Saudi Arabia

    G. M. Bahaa

  3. Faculty of Natural and Agricultural Sciences, Institute of Groundwater Studies, University of Free State, 9301, Bloemfontein, South Africa

    A. Atangana

Authors
  1. G. M. Bahaa
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. A. Atangana
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to G. M. Bahaa .

Editor information

Editors and Affiliations

  1. CONACYT-Tecnológico Nacional de México, Centro Nacional de Investigación y Desarrollo Tecnológico, Cuernavaca, Morelos, Mexico

    José Francisco Gómez

  2. CONACYT-Instituto de Ingeniería, Universidad Nacional Autónoma de México, Mexico City, Mexico

    Lizeth Torres

  3. Tecnológico Nacional de México, Centro Nacional de Investigación y Desarrollo Tecnológico, Cuernavaca, Morelos, Mexico

    Ricardo Fabricio Escobar

Rights and permissions

Reprints and permissions

Copyright information

© 2019 Springer Nature Switzerland AG

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

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)

Publish with us

Policies and ethics

Search

Navigation