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Pontryagin Maximum Principle for Distributed-Order Fractional Systems

Author

Listed:
  • Faïçal Ndaïrou

    (Center for Research and Development in Mathematics and Applications (CIDMA), Department of Mathematics, University of Aveiro, 3810-193 Aveiro, Portugal
    This research is part of first author’s Ph.D. project, which is carried out at the University of Aveiro under the Doctoral Program in Applied Mathematics of Universities of Minho, Aveiro, and Porto (MAP-PDMA).
    These authors contributed equally to this work.)

  • Delfim F. M. Torres

    (Center for Research and Development in Mathematics and Applications (CIDMA), Department of Mathematics, University of Aveiro, 3810-193 Aveiro, Portugal
    These authors contributed equally to this work.)

Abstract

We consider distributed-order non-local fractional optimal control problems with controls taking values on a closed set and prove a strong necessary optimality condition of Pontryagin type. The possibility that admissible controls are subject to pointwise constraints is new and requires more sophisticated techniques to include a maximality condition. We start by proving results on continuity of solutions due to needle-like control perturbations. Then, we derive a differentiability result on the state solutions with respect to the perturbed trajectories. We end by stating and proving the Pontryagin maximum principle for distributed-order fractional optimal control problems, illustrating its applicability with an example.

Suggested Citation

  • Faïçal Ndaïrou & Delfim F. M. Torres, 2021. "Pontryagin Maximum Principle for Distributed-Order Fractional Systems," Mathematics, MDPI, vol. 9(16), pages 1-12, August.
    • Handle: RePEc:gam:jmathe:v:9:y:2021:i:16:p:1883-:d:610461
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    References listed on IDEAS

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    1. Yonatan Golan & Eilon Sherman, 2017. "Resolving mixed mechanisms of protein subdiffusion at the T cell plasma membrane," Nature Communications, Nature, vol. 8(1), pages 1-15, August.
    2. Kumar, Yashveer & Singh, Vineet Kumar, 2021. "Computational approach based on wavelets for financial mathematical model governed by distributed order fractional differential equation," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 190(C), pages 531-569.
    3. Almeida, Ricardo & Morgado, M. Luísa, 2018. "The Euler–Lagrange and Legendre equations for functionals involving distributed–order fractional derivatives," Applied Mathematics and Computation, Elsevier, vol. 331(C), pages 394-403.
    4. M. A. Abdelkawy, 2021. "Numerical solutions for fractional initial value problems of distributed-order," International Journal of Modern Physics C (IJMPC), World Scientific Publishing Co. Pte. Ltd., vol. 32(07), pages 1-13, July.
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