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

  • Jean-Daniel Djida
  • Gisèle Mophou
  • Iván AreaEmail author


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.


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

Mathematics Subject Classification

49J20 49K20 26A33 



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.


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Copyright information

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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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