Optimal Control of Diffusion Equation with Fractional Time Derivative with Nonlocal and Nonsingular Mittag-Leffler Kernel

Article

Abstract

In this paper, we consider a diffusion equation with fractional time derivative with nonsingular Mittag-Leffler kernel in Hilbert spaces. We first prove the existence and uniqueness of solution by means of a spectral argument. Then, we consider a distributed controlled fractional diffusion problem. We show that there exists a unique optimal control, which can act on the system in order to approach the state of the system by a given state at minimal cost. Finally, using the Euler–Lagrange first-order optimality condition, we obtain an optimality system, which characterizes the optimal control.

Keywords

Mittag-Leffler functions Time-fractional differential equation Optimality system Euler–Lagrange optimality conditions 

Mathematics Subject Classification

49J20 49K20 26A33 

Notes

Acknowledgements

The first author is grateful for the facilities provided by the German research Chairs and the Teacher Training Program of AIMS-Cameroon. The first author is also indebted to the AIMS-Cameroon 2017–2018 Tutor fellowship. The second author was supported by the Alexander von Humboldt foundation, under the program financed by the BMBF entitled “German research Chairs”. The work of the third author has been partially supported by the Agencia Estatal de Innovación (AEI) of Spain under Grant MTM2016-75140-P, cofinanced by the European Community fund FEDER, and Xunta de Galicia, Grant R 2016/022. Besides the authors are grateful to the unknown referees for their valuable suggestions.

References

  1. 1.
    Metzler, R., Klafter, J.: The random walk’s guide to anomalous diffusion: a fractional dynamics approach. Phys. Rep. 339(1), 1–77 (2000)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Schneider, W.R., Wyss, W.: Fractional diffusion and wave equations. J. Math. Phys. 30(1), 134–144 (1989)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Lei, S.-L., Sun, H.-W.: A circulant preconditioner for fractional diffusion equations. J. Comput. Phys. 242, 715–725 (2013)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Wang, H., Wang, K., Sircar, T.: A direct \(O(N\log ^2N)\) finite difference method for fractional diffusion equations. J. Comput. Phys. 229, 8095–8104 (2010)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Mophou, G.: Optimal control of fractional diffusion equation. Comput. Math. Appl. 61, 68–78 (2011)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Sakamoto, K., Yamamoto, M.: Initial value/boundary value problems for fractional diffusion-wave equations and applications to some inverse problems. J. Math. Anal. Appl. 382, 426–447 (2011)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Choi, J.H., Rim, H., Sakthivel, R.: On certain exact solutions of diffusive predator-prey system of fractional order. Chin. J. Phys. 54, 135–146 (2016)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Agrawal, O.P.: A general formulation and solution scheme for fractional optimal control problems. Nonlinear Dyn. 38, 323–337 (2004)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Agrawal, O.P.: Formulation of Euler–Lagrange equations for fractional variational problems. J. Math. Anal. 272, 368–379 (2002)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Frederico, G.S.F., Torres, D.F.M.: Fractional optimal control in the sense of caputo and the fractional Noether’s Theorem. Int. Math. Forum. 3(10), 479–493 (2008)MathSciNetMATHGoogle Scholar
  11. 11.
    Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam (2006)MATHGoogle Scholar
  12. 12.
    Samko, S., Kilbas, A.A., Marichev, O.: Fractional Integrals and Derivatives. Taylor & Francis, London (1993)MATHGoogle Scholar
  13. 13.
    Jin, B., Lazarov, R., Zhou, Z.: Error estimates for a semi-discrete finite element method for fractional order parabolic equations. SIAM J. Numer. Anal. 51(1), 445–466 (2013)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Atangana, A., Baleanu, D.: New fractional derivatives with non-local and non-singular kernel: theory and application to heat transfer model. Thermal Sci. 20(2), 763–769 (2016)CrossRefGoogle Scholar
  15. 15.
    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)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    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)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Podlubny, I.: Fractional Differential Equations. Academic Press, San Diego (1999)MATHGoogle Scholar
  18. 18.
    Bazhlekova E.: Fractional Evolution Equation in Banach Spaces. Ph.D. Thesis. Eindhoven University of Technology (2001)Google Scholar
  19. 19.
    Diethelm, K., Luchko, Y.: Numerical solution of linear multi-term initial value problems of fractional order. J. Comput. Anal. Appl. 6, 243–263 (2004)MathSciNetMATHGoogle Scholar
  20. 20.
    Djida J.D., Atangana A., and Area I.: Parabolic problem with fractional time derivative with nonlocal and nonsingular Mittag-Leffler kernel. arXiv:1701.01479 (2017)
  21. 21.
    Baleanu, D., Joseph, C., Mophou, G.: Low regret control for a fractional wave equation with incomplete data. Adv. Differ. Equ. 1, 240 (2016)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Departamento de Estatística, Análise Matemática e OptimizaciónUniversidade de Santiago de CompostelaSantiago de CompostelaSpain
  2. 2.African Institute for Mathematical Sciences (AIMS)LimbeCameroon
  3. 3.Laboratoire L.A.M.I.A., Département de Mathématiques et InformatiqueUniversité des AntillesPointe-à-PitreFrance
  4. 4.Departamento de Matemática Aplicada II, E.E. Aeronáutica e do EspazoUniversidade de VigoOurenseSpain

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