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Optimal control problem for variable-order fractional differential systems with time delay involving Atangana–Baleanu derivatives

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  • Bahaa, G.M.

Abstract

The Optimal Control Problem (OCP) for variable-order fractional differential systems with time delay is considered. The fractional time derivative is Atangana–Baleanu derivatives in a Caputo sense. The existence and the uniqueness results are derived. Then we show that the considered optimal control problem has a unique solution. The performance index of a (FOCP) is considered as an integral function of both state and control variables, and the dynamic constraints are expressed by a Partial Fractional Differential Equation (PFDE) with variable-order and time delay. The time horizon is fixed. Finally, we impose some constraints on the boundary control. Interpreting the Euler-Lagrange first order optimality condition with an adjoint problem defined by means of right fractional Atangana–Baleanu derivative, we obtain an optimality system for the optimal control. To obtain the optimality conditions for the given problem, the generalization of the Dubovitskii–Milyutin Theorem was applied. To illustrate the results we introduce some examples.

Suggested Citation

  • Bahaa, G.M., 2019. "Optimal control problem for variable-order fractional differential systems with time delay involving Atangana–Baleanu derivatives," Chaos, Solitons & Fractals, Elsevier, vol. 122(C), pages 129-142.
    • Handle: RePEc:eee:chsofr:v:122:y:2019:i:c:p:129-142
    DOI: 10.1016/j.chaos.2019.03.001
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    1. Mophou, G. & Tao, S. & Joseph, C., 2015. "Initial value/boundary value problem for composite fractional relaxation equation," Applied Mathematics and Computation, Elsevier, vol. 257(C), pages 134-144.
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    Cited by:

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    2. Hasib Khan & Jehad Alzabut & Haseena Gulzar & Osman Tunç & Sandra Pinelas, 2023. "On System of Variable Order Nonlinear p-Laplacian Fractional Differential Equations with Biological Application," Mathematics, MDPI, vol. 11(8), pages 1-17, April.
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    4. Panda, Sumati Kumari & Abdeljawad, Thabet & Ravichandran, C., 2020. "A complex valued approach to the solutions of Riemann-Liouville integral, Atangana-Baleanu integral operator and non-linear Telegraph equation via fixed point method," Chaos, Solitons & Fractals, Elsevier, vol. 130(C).
    5. Shojaeizadeh, T. & Mahmoudi, M. & Darehmiraki, M., 2021. "Optimal control problem of advection-diffusion-reaction equation of kind fractal-fractional applying shifted Jacobi polynomials," Chaos, Solitons & Fractals, Elsevier, vol. 143(C).
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    7. Heydari, M. H. & Atangana, A., 2020. "An optimization method based on the generalized Lucas polynomials for variable-order space-time fractional mobile-immobile advection-dispersion equation involving derivatives with non-singular kernels," Chaos, Solitons & Fractals, Elsevier, vol. 132(C).

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