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On the Existence and Continuous Dependence on Parameter of Solutions to Some Fractional Dirichlet Problem with Application to Lagrange Optimal Control Problem

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  • Rafał Kamocki

    (University of Lodz)

  • Marek Majewski

    (University of Lodz)

Abstract

In the paper, a Lagrange optimal control problem governed by a fractional Dirichlet problem with the Riemann–Liouville derivative is considered. To begin with, based on some variational method, the existence and continuous dependence of solution to the aforementioned Dirichlet problem is investigated. Then, continuous dependence is applied to show the existence of optimal solution to the Lagrange problem. An important point is that the solution to Dirichlet problem does need to be unique; therefore, the above dependence should be understood as a continuity of some multifunction—the concept of the Kuratowski–Painlevé limit of the sequence of sets is used to formulate this property.

Suggested Citation

  • Rafał Kamocki & Marek Majewski, 2017. "On the Existence and Continuous Dependence on Parameter of Solutions to Some Fractional Dirichlet Problem with Application to Lagrange Optimal Control Problem," Journal of Optimization Theory and Applications, Springer, vol. 174(1), pages 32-46, July.
    • Handle: RePEc:spr:joptap:v:174:y:2017:i:1:d:10.1007_s10957-016-0954-6
    DOI: 10.1007/s10957-016-0954-6
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    References listed on IDEAS

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    1. Tian Liang Guo, 2013. "The Necessary Conditions of Fractional Optimal Control in the Sense of Caputo," Journal of Optimization Theory and Applications, Springer, vol. 156(1), pages 115-126, January.
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    3. Blaszczyk, Tomasz & Ciesielski, Mariusz, 2015. "Fractional oscillator equation – Transformation into integral equation and numerical solution," Applied Mathematics and Computation, Elsevier, vol. 257(C), pages 428-435.
    4. Shkelzen Cakaj (ed.), 2010. "Modeling Simulation and Optimization - Tolerance and Optimal Control," Books, IntechOpen, number 707, January-J.
    5. JinRong Wang & Michal Fec̆kan & Yong Zhou, 2013. "Relaxed Controls for Nonlinear Fractional Impulsive Evolution Equations," Journal of Optimization Theory and Applications, Springer, vol. 156(1), pages 13-32, January.
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