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Optimality conditions and a solution scheme for fractional optimal control problems

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Abstract

We formulate necessary conditions for optimality in Optimal control problems with dynamics described by differential equations of fractional order (derivatives of arbitrary real order). Then by using an expansion formula for fractional derivative, optimality conditions and a new solution scheme is proposed. We assumed that the highest derivative in the differential equation of the process is of integer order. Two examples are treated in detail.

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References

  • Agrawal OP (2002) Formulation of Euler-Lagrange equations for fractional variational problems. J Math Anal Appl 272:368–379

    Article  MATH  MathSciNet  Google Scholar 

  • Agrawal OP (2004) A general formulation and solution for fractional optimal control problems. Nonlinear Dyn 38:323–337

    Article  MATH  Google Scholar 

  • Agrawal OP (2006) Fractional variational calculus and the transversality conditions. J Phys A Math Gen 39:10375–10384

    Article  MATH  Google Scholar 

  • Agrawal OP (2008) A genaral finite element formulatiuon for fractional variational problems. J Math Anal Appl 337:1–12

    Article  MATH  MathSciNet  Google Scholar 

  • Atanackovic TM, Stankovic B (2004) An expansion formula for fractional derivatives and its applications. Fractional Calculus and Applied Analysis 7(3):365–378

    MATH  MathSciNet  Google Scholar 

  • Atanackovic TM, Stankovic B (2007a) On a class of differential equations with left and right fractional derivatives. ZAMM, Z Angew Math Mech 87:537–546

    Article  MATH  MathSciNet  Google Scholar 

  • Atanackovic TM, Stankovic B (2007b) On a differential equation with left and right fractional derivatives. Fractional Calculus Applied Analysis 10:138–150

    MathSciNet  Google Scholar 

  • Atanackovic TM, Konjik S, Pilipovic S (2008) Variational problems with fractional derivatives: Euler–Lagrange equations. J Phys A Math Theor 41. doi:10.1088/1751-8113/41/9/095201

  • Baleanu D, Akvar T (2004) Lagragians with linear velocities within Riemann–Liouville fractional derivatives. Nuovo Cimento B 119:73–79

    Google Scholar 

  • Jelicic ZD, Petrovacki ND (2007) Fractional derivative model of EDFA with ASE. In: SIAM conference on control and its applications, San Francisco June 2007

  • Kilbas AA, Srivastava HM, Trujillo JJ (2006) Theory and applications of fractional differential equations. Elsevier, North-Holland Mathematics Studies 204

    MATH  Google Scholar 

  • Petrovacki N, Jelicic ZD (2006) Optimal transient response of erbium-doped fiber amplifiers. In: IEEE international conference on numerical simulation of semiconductor optoelectronic devices. NUSOD ’06, Singapore, Sept 2006

  • Rabei EM, Nawafleh KI, Hijjawi RS, Muslih SI, Baleanu D (2007) The Hamilton formalism with fractional derivatives. J Math Anal Appl 327:891–897

    Article  MATH  MathSciNet  Google Scholar 

  • Rekhviashvili S Sh (2004) The lagrange formalism with fractional derivatives in problems of mechanics. Tech Phys Lett 30:33–37

    Article  Google Scholar 

  • Ruge P, Trinks C (2004) Consistent modelling of infinite beams by fractional dynamics. Nonlinear Dyn 38:267–284

    Article  MATH  Google Scholar 

  • Samko SG, Kilbas AA, Marichev OI (1993) Fractional integrals and derivatives. Gordon and Breach Science Publishers

  • Vujanovic BD, Atanackovic TM (2004) An introduction to modern variational principles of mechanics. Birkhäuser, Boston

    Google Scholar 

  • Yuan L, Agrawal OP (1998) A numerical scheme for dynamic systems containing fractional derivatives. In: Proceedings of DETC’98 1998 ASME design engineering technical conferences, Atlanta Georgia September 13–16

  • Yuan L, Agrawal OP (2002) A numerical scheme for dynamic systems containing fractional derivatives. J Vib Acoust 124:321–324

    Article  Google Scholar 

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Authors and Affiliations

  1. Faculty of Technical Sciences, University of Novi Sad, Trg D. Obradovica 6, 21000, Novi Sad, Serbia

    Zoran D. Jelicic

  2. Mccrometer Inc, 3255 W. Stetson Avenue, Hemet, CA, 92545, USA

    Nebojsa Petrovacki

Authors
  1. Zoran D. Jelicic
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  2. Nebojsa Petrovacki
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Correspondence to Zoran D. Jelicic.

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Jelicic, Z.D., Petrovacki, N. Optimality conditions and a solution scheme for fractional optimal control problems. Struct Multidisc Optim 38, 571–581 (2009). https://doi.org/10.1007/s00158-008-0307-7

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  • DOI: https://doi.org/10.1007/s00158-008-0307-7

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