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Volume 4 Issue 2
Apr.  2017

IEEE/CAA Journal of Automatica Sinica

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Article Navigation >  IEEE/CAA Journal of Automatica Sinica  > 2017  >  4(2): 340-352
Matheus J. Lazo and DelfimF.M. Torres, "Variational Calculus With Conformable Fractional Derivatives," IEEE/CAA J. Autom. Sinica, vol. 4, no. 2, pp. 340-352, Apr. 2017. doi: 10.1109/JAS.2016.7510160
Citation: Matheus J. Lazo and DelfimF.M. Torres, "Variational Calculus With Conformable Fractional Derivatives," IEEE/CAA J. Autom. Sinica, vol. 4, no. 2, pp. 340-352, Apr. 2017. doi: 10.1109/JAS.2016.7510160
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Variational Calculus With Conformable Fractional Derivatives

doi: 10.1109/JAS.2016.7510160
  • 1.

    Instituto de Matemática, Estatística e Física-FURG, Rio Grande, RS 96.201-900, Brazil

  • 2.

    Center for Research and Development in Mathematics and Applications (CIDMA), Department of Mathematics, University of Aveiro, Aveiro 3810-193, Portugal

Funds:  This work was partially supported by CNPq and CAPES (Brazilian research funding agencies), and Portuguese funds through the Center for Research and Development in Mathematics and Applications (CIDMA), and also the Portuguese Foundation for Science and Technology (FCT), within project UID/MAT/04106/2013
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    Abstract
  • Invariant conditions for conformable fractional problems of the calculus of variations under the presence of external forces in the dynamics are studied. Depending on the type of transformations considered, different necessary conditions of invariance are obtained. As particular cases, we prove fractional versions of Noether's symmetry theorem. Invariant conditions for fractional optimal control problems, using the Hamiltonian formalism, are also investigated. As an example of potential application in Physics, we show that with conformable derivatives it is possible to formulate an Action Principle for particles under frictional forces that is far simpler than the one obtained with classical fractional derivatives.

     

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  • [1]
    A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Volume 204 (North-Holland Mathematics Studies). Amsterdam: Elsevier Science, 2006.
    [2]
    K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations. New York: Wiley-Interscience, 1993.
    [3]
    I. Podlubny, Fractional Differential Equations, Mathematics in Science and Engineering. San Diego, CA: Academic Press, Inc., 1999.
    [4]
    J. T. Machado, V. Kiryakova, and F. Mainardi, "Recent history of fractional calculus, " Commun. Nonlinear Sci. Numer. Simul. , vol. 16, no. 3, pp. 1140-1153, Mar. 2011. http://www.sciencedirect.com/science/article/pii/S1007570410003205
    [5]
    R. Khalil, M. Al Horani, A. Yousef, and M. Sababheh, "A new definition of fractional derivative, " J. Comput. Appl. Math. , vol. 264, pp. 65-70, Jul. 2014. http://www.sciencedirect.com/science/article/pii/S0377042714000065
    [6]
    T. Abdeljawad, "On conformable fractional calculus, " J. Comput. Appl. Math. , vol. 279, pp. 57-66, May2015. http://www.sciencedirect.com/science/article/pii/S0377042714004622
    [7]
    D. R. Anderson and R. I. Avery, "Fractional-order boundary value problem with Sturm-Liouville boundary conditions, " Electron. J. Diff. Equ. , vol. 2015, no. 29, pp. 10, Jan. 2015. http://www.wenkuxiazai.com/doc/6d8cb0076c85ec3a87c2c5c2.html
    [8]
    H. Batarfi, J. Losada, J. J. Nieto, and W. Shammakh, "Three-point boundary value problems for conformable fractional differential equations, " J. Funct. Spaces, vol. 2015, Article ID 706383, 2015. http://www.sciencedirect.com/science/article/pii/S0898122103000981
    [9]
    B. Bayour and D. F. M. Torres, "Existence of solution to a local fractional nonlinear differential equation, " J. Comput. Appl. Math. , vol. 312, pp. 127--133, Mar. 2017.
    [10]
    N. Benkhettou, S. Hassani, and D. F. M. Torres, "A conformable fractional calculus on arbitrary time scales, " J. King Saud Univ. Sci. , vol. 28, no. 1, pp. 93-98, Jan. 2016. http://www.sciencedirect.com/science/article/pii/S1018364715000464
    [11]
    F. Riewe, "Mechanics with fractional derivatives, " Phys. Rev. E, vol. 55, no. 3, pp. 3581-3592, Mar. 1997. http://adsabs.harvard.edu/abs/1997PhRvE..55.3581R
    [12]
    P. S. Bauer, "Dissipative dynamical systems. I, " Proc. Natl. Acad. Sci. USA, vol. 17, pp. 311-314, Jun. 1931. doi: 10.1007%2FBF00276493
    [13]
    O. P. Agrawal, "Formulation of Euler-Lagrange equations for fractional variational problems, " J. Math. Anal. Appl. , vol. 272, no. 1, pp. 368-379, Aug. 2002. http://www.sciencedirect.com/science/article/pii/S0022247X02001804
    [14]
    R. Almeida and D. F. M. Torres, "Necessary and sufficient conditions for the fractional calculus of variations with Caputo derivatives, " Commun. Nonlinear Sci. Numer. Simul. , vol. 16, no. 3, pp. 1490-1500, 2011. http://philosophy.wisc.edu/hausman/341/Skill/nec-suf.htm
    [15]
    D. Baleanu and O. P. Agrawal, "Fractional Hamilton formalism within Caputo's derivative, " Czechoslovak J. Phys. , vol. 56, no. 10-11, pp. 1087-1092, Oct. 2006. doi: 10.1007/s10582-006-0406-x
    [16]
    J. Cresson, "Fractional embedding of differential operators and Lagrangian systems, " J. Math. Phys. , vol. 48, no. 3, pp. 033504, Mar. 2007. doi: 10.1063/1.2483292
    [17]
    M. J. Lazo and D. F. M. Torres, "The DuBois-Reymond fundamental lemma of the fractional calculus of variations and an Euler-Lagrange equation involving only derivatives of Caputo, " J. Optim. Theory Appl. , vol. 156, no. 1, pp. 56-67, Jan. 2013. doi: 10.1007/s10957-012-0203-6
    [18]
    T. Odzijewicz, A. B. Malinowska, and D. F. M. Torres, "Fractional variational calculus with classical and combined Caputo derivatives, " Nonlinear Anal. , vol. 75, no. 3, pp. 1507-1515, Feb. 2012. http://www.sciencedirect.com/science/article/pii/S0362546X11000113
    [19]
    T. Odzijewicz, A. B. Malinowska, and D. F. M. Torres, "Fractional calculus of variations in terms of a generalized fractional integral with applications to physics, " Abstr. Appl. Anal. , vol. 2012, Article ID 871912, 2012. https://www.hindawi.com/journals/aaa/2012/871912/
    [20]
    R. Almeida, S. Pooseh, and D. F. M. Torres, Computational Methods in the Fractional Calculus of Variations. London: Imperial College Press, 2015.
    [21]
    A. B. Malinowska, T. Odzijewicz, and D. F. M. Torres, Advanced Methods in the Fractional Calculus of Variations. New York: Springer International Publishing, 2015.
    [22]
    A. B. Malinowska and D. F. M. Torres, Introduction to the Fractional Calculus of Variations. London: Imperial College Press, 2012.
    [23]
    M. J. Lazo and C. E. Krumreich, "The action principle for dissipative systems, " J. Math. Phys. , vol. 55, no. 12, pp. 122902, 2014. doi: 10.1063/1.4903991
    [24]
    J. D. Logan, Invariant Variational Principles. Vol.138. Mathematics in Science and Engineering. New York, San Francisco, Lindon: Academic Press, 1977.
    [25]
    D. F. M. Torres, "Proper extensions of Noether's symmetry theorem for nonsmooth extremals of the calculus of variations, " Commun. Pure Appl. Anal. , vol. 3, no. 3, pp. 491-500, Sep. 2004. https://www.researchgate.net/publication/280005301_Proper_extensions_of_Noether%27s_symmetry_theorem_for_nonsmooth_extremals_of_the_calculus_of_variations
    [26]
    G. S. F. Frederico and D. F. M. Torres, "Non-conservative Noether's theorem for fractional action-like variational problems with intrinsic and observer times, " Int. J. Ecol. Econ. Stat. , vol. 9, no. F07, pp. 74-82, Nov. 2007. http://www.academia.edu/2372009/Non-conservative_Noethers_theorem_for_fractional_action-like_variational_problems_with_intrinsic_and_observer_times
    [27]
    G. S. F. Frederico and D. F. M. Torres, "A formulation of Noether's theorem for fractional problems of the calculus of variations, " J. Math. Anal. Appl. , vol. 334, no. 2, pp. 834-846, Oct. 2007. http://www.sciencedirect.com/science/article/pii/S0022247X07000340
    [28]
    G. S. F. Frederico and D. F. M. Torres, "Fractional conservation laws in optimal control theory, " Nonlinear Dyn. , vol. 53, no. 3, pp. 215-222, Aug. 2008. doi: 10.1007/s11071-007-9309-z
    [29]
    G. S. F. Frederico and D. F. M. Torres, "Fractional optimal control in the sense of Caputo and the fractional Noether's theorem, " Int. Math. Forum, vol. 3, no. 10, pp. 479-493, Sep. 2008. http://www.wenkuxiazai.com/doc/e6c1f7fec8d376eeaeaa31f2-2.html
    [30]
    G. S. F. Frederico and D. F. M. Torres, "Fractional Noether's theorem in the Riesz-Caputo sense, " Appl. Math. Comput. , vol. 217, no. 3, pp. 1023-1033, Oct. 2010. http://www.sciencedirect.com/science/article/pii/S0096300310001244
    [31]
    N. Benkhettou, A. M. C. Brito Da Cruz, and D. F. M. Torres, "A fractional calculus on arbitrary time scales: Fractional differentiation and fractional integration, " Signal Proc. , vol. 107, pp. 230-237, Feb. 2015. http://www.sciencedirect.com/science/article/pii/S0165168414002448
    [32]
    N. Benkhettou, A. M. C. Brito da Cruz, and D. F. M. Torres, "Nonsymmetric and symmetric fractional calculi on arbitrary nonempty closed sets, " Math. Methods Appl. Sci. , vol. 39, no. 2, pp. 261-279, Jan. 2016. http://www.oalib.com/paper/3941045
    [33]
    W. Sarlet and F. Cantrijn, "Generalizations of Noether's theorem in classical mechanics, " SIAM Rev. , vol. 23, no. 4, pp. 467-494, 1981. doi: 10.1137/1023098
    [34]
    G. S. F. Frederico and D. F. M. Torres, "Nonconservative Noether's theorem in optimal control, " Int. J. Tomogr. Stat. , vol. 5, no. W07, pp. 109-114, 2007. http://www.academia.edu/2372000/Nonconservative_Noethers_theorem_in_optimal_control
    [35]
    D. F. M. Torres, "On the Noether theorem for optimal control, " Eur. J. Control, vol. 8, no. 1, pp. 56-63, 2002.
    [36]
    D. F. M. Torres, "Conservation laws in optimal control, " in Dynamics, Bifurcations, and Control, vol. 273, F. Colonius and L Grüne, Eds. Berlin: Springer, 2002, pp. 287-296.
    [37]
    D. F. M. Torres, "Quasi-invariant optimal control problems, " Port. Math. , vol. 61, no. 1, pp. 97-114, 2004. https://archive.org/details/arxiv-math0302264
    [38]
    S. Pooseh, R. Almeida, and D. F. M. Torres, "Fractional order optimal control problems with free terminal time, " J. Ind. Manag. Optim. , vol. 10, no. 2, pp. 363-381, Apr. 2014. doi: 10.1186/s13662-016-0976-2
    [39]
    D. S. Dukić, "Noether's theorem for optimum control systems, " Int. J. Control, vol. 18, no. 3, pp. 667-672, Jul. 1973. doi: 10.1080/00207177308932544
    [40]
    A. S. Balankin, J. Bory-Reyes, and M. Shapiro, "Towards a physics on fractals: differential vector calculus in three-dimensional continuum with fractal metric, " Phys. A, vol. 444, pp. 345-359, Feb. 2016. http://www.sciencedirect.com/science/article/pii/S037843711500881X
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