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Euler–Lagrange Equations for Lagrangians Containing Complex-order Fractional Derivatives

Author

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  • Teodor M. Atanacković

    (University of Novi Sad)

  • Marko Janev

    (Serbian Academy of Arts and Sciences)

  • Stevan Pilipović

    (University of Novi Sad)

  • Dušan Zorica

    (Serbian Academy of Arts and Sciences
    University of Novi Sad)

Abstract

Two variational problems of finding the Euler–Lagrange equations corresponding to Lagrangians containing fractional derivatives of real- and complex-order are considered. The first one is the unconstrained variational problem, while the second one is the fractional optimal control problem. The expansion formula for fractional derivatives of complex-order is derived in order to approximate the fractional derivative appearing in the Lagrangian. As a consequence, a sequence of approximated Euler–Lagrange equations is obtained. It is shown that the sequence of approximated Euler–Lagrange equations converges to the original one in the weak sense as well as that the sequence of the minimal values of approximated action integrals tends to the minimal value of the original one.

Suggested Citation

  • Teodor M. Atanacković & Marko Janev & Stevan Pilipović & Dušan Zorica, 2017. "Euler–Lagrange Equations for Lagrangians Containing Complex-order Fractional Derivatives," Journal of Optimization Theory and Applications, Springer, vol. 174(1), pages 256-275, July.
    • Handle: RePEc:spr:joptap:v:174:y:2017:i:1:d:10.1007_s10957-016-0873-6
    DOI: 10.1007/s10957-016-0873-6
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    References listed on IDEAS

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    1. Ali Lotfi & Sohrab Ali Yousefi, 2014. "Epsilon-Ritz Method for Solving a Class of Fractional Constrained Optimization Problems," Journal of Optimization Theory and Applications, Springer, vol. 163(3), pages 884-899, December.
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    Cited by:

    1. Atanacković, Teodor M. & Janev, Marko & Pilipović, Stevan, 2018. "Non-linear boundary value problems involving Caputo derivatives of complex fractional order," Applied Mathematics and Computation, Elsevier, vol. 334(C), pages 326-342.

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