A New Approach for Solving Optimal Control Problem by Using Orthogonal Function

  • Akram KheirabadiEmail author
  • Asadollah Mahmoudzadeh Vaziri
  • Sohrab Effati
Conference paper
Part of the Lecture Notes on Multidisciplinary Industrial Engineering book series (LNMUINEN)


In the present paper we introduce a numerical technique for solving fractional optimal control problems (FOCP) based on an orthonormal wavelet. First we approximate the involved functions by Sine-Cosine wavelet basis; then, an operational matrix is used to transfer the given problem in to a linear system of algebraic equations. In fact operational matrix of the Riemann-Liouville fractional integration and derivative of Sine-Cosine wavelet are employed to achieve a linear algebraic equation, in place of the dynamical system in terms of the unknown coefficients. The solution of this system, gives us the solution of original problem. A numerical example is also given.


Fractional optimal control problem Sine-Cosine wavelet Operational matrix Caputo derivative Riemann-Liouville fractional integration 


  1. 1.
    Agrawal OP (2008) A formulation and numerical scheme for fractional optimal control problems. J Vibr Control 14(9–10):1291–1299MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Bagley RL, Torvik P (1983) A theoretical basis for the application of fractional calculus to viscoelasticity. J Rheol 27(3):201–210CrossRefzbMATHGoogle Scholar
  3. 3.
    Baillie RT (1996) Long memory processes and fractional integration in econometrics. J Econometrics 73(1):5–59MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Bohannan GW (2008) Analog fractional order controller in temperature and motor control applications. J Vibr Control 14(9–10):1487–1498MathSciNetCrossRefGoogle Scholar
  5. 5.
    Canuto C, Hussaini MY, Quarteroni A, Zang TA Jr (1988) Spectral methods in fluid dynamics. Annu Rev Fluid Mech 57(196):339–367Google Scholar
  6. 6.
    Dehghan M, Manafian J, Saadatmandi A (2010) The solution of the linear fractional partial differential equations using the homotopy analysis method. Z Naturforsch A 65(Z. Naturforsch):935–949Google Scholar
  7. 7.
    Dehghan M, Manafian J, Saadatmandi A (2010) Solving nonlinear fractional partial differential equations using the homotopy analysis method. Numer Methods Partial Differ Equ 26(2):448–479MathSciNetzbMATHGoogle Scholar
  8. 8.
    He J (1998) Nonlinear oscillation with fractional derivative and its applications. In: International conference on vibrating engineering, Dalian, China, vol 98, pp 288–291Google Scholar
  9. 9.
    He J (1999) Some applications of nonlinear fractional differential equations and their approximations. Bull Sci Technol 15(2):86–90MathSciNetGoogle Scholar
  10. 10.
    Jafari H, Tajadodi H (2014) Fractional order optimal control problems via the operational matrices of bernstein polynomials. Upb Sci Bull 76(3):115–128MathSciNetzbMATHGoogle Scholar
  11. 11.
    Kajani MT, Ghasemi M, Babolian E (2006) Numerical solution of linear integro-differential equation by using sine-cosine wavelets. Appl Math Comput 180(2):569–574MathSciNetzbMATHGoogle Scholar
  12. 12.
    Li Y, Sun N (2011) Numerical solution of fractional differential equations using the generalized block pulse operational matrix. Comput Math Appl 62(3):1046–1054MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Liu F, Agrawal OP et al (1999) Fractional differential equations: an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications. Academic PressGoogle Scholar
  14. 14.
    Lotfi A, Dehghan M, Yousefi SA (2011) A numerical technique for solving fractional optimal control problems. Comput Math Appl 62(3):1055–1067MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Momani S, Odibat Z (2007) Numerical comparison of methods for solving linear differential equations of fractional order. Chaos Solitons Fractals 31(5):1248–1255MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Odibat ZM, Shawagfeh NT (2007) Generalized taylor’s formula. Appl Math Comput 186(1):286–293MathSciNetzbMATHGoogle Scholar
  17. 17.
    Panda R, Dash M (2006) Fractional generalized splines and signal processing. Sig Process 86(9):2340–2350CrossRefzbMATHGoogle Scholar
  18. 18.
    Rossikhin YA, Shitikova MV (1997) Applications of fractional calculus to dynamic problems of linear and nonlinear hereditary mechanics of solids. Appl Mech Rev 50:15–67CrossRefGoogle Scholar
  19. 19.
    Saadatmandi A, Dehghan M (2010) A new operational matrix for solving fractional-order differential equations. Comput Math Appl 59(3):1326–1336MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Saez D (2009) Analytical solution of a fractional diffusion equation by variational iteration method. Comput Math Appl 57(3):483–487MathSciNetCrossRefGoogle Scholar
  21. 21.
    Sen S (2011) Fractional optimal control problems: A pseudo-state space approach. J Vibr Control 17(17):1034–1041MathSciNetzbMATHGoogle Scholar
  22. 22.
    Shawagfeh NT (2002) Analytical approximate solutions for nonlinear fractional differential equations. Appl Math Comput 131(2–3):517–529MathSciNetzbMATHGoogle Scholar
  23. 23.
    Sohrabi S (2011) Comparison chebyshev wavelets method with bpfs method for solving abel’s integral equation. Ain Shams Eng J 2(3–4):249–254CrossRefGoogle Scholar
  24. 24.
    Sweilam NH, Alajami TM, Hoppe RHW (2013) Numerical solution of some types of fractional optimal control problems. Sci World J 2013(2):306237Google Scholar
  25. 25.
    Tangpong XW, Agrawal OP (2009) Fractional optimal control of continuum systems. J Vibr Acoust 131(2):557–557CrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG 2018

Authors and Affiliations

  • Akram Kheirabadi
    • 1
    Email author
  • Asadollah Mahmoudzadeh Vaziri
    • 1
  • Sohrab Effati
    • 2
  1. 1.Department of Mathematics, Faculty of Mathematical Science and StatisticsUniversity of BirjandBirjandIran
  2. 2.Department of Applied Mathematics, Faculty of Mathematical Science and StatisticsFerdowsi University of MashhadMashhadIran

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