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Epsilon-Ritz Method for Solving a Class of Fractional Constrained Optimization Problems

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Abstract

In this paper, epsilon and Ritz methods are applied for solving a general class of fractional constrained optimization problems. The goal is to minimize a functional subject to a number of constraints. The functional and constraints can have multiple dependent variables, multiorder fractional derivatives, and a group of initial and boundary conditions. The fractional derivative in the problem is in the Caputo sense. The constrained optimization problems include isoperimetric fractional variational problems (IFVPs) and fractional optimal control problems (FOCPs). In the presented approach, first by implementing epsilon method, we transform the given constrained optimization problem into an unconstrained problem, then by applying Ritz method and polynomial basis functions, we reduce the optimization problem to the problem of optimizing a real value function. The choice of polynomial basis functions provides the method with such a flexibility that initial and boundary conditions can be easily imposed. The convergence of the method is analytically studied and some illustrative examples including IFVPs and FOCPs are presented to demonstrate validity and applicability of the new technique.

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References

  1. Agrawal, O.M.P.: Formulation of Euler-Lagrange equations for fractional variational problems. J. Math. Anal. Appl. 272, 368–379 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  2. Agrawal, O.M.P.: A general formulation and solution scheme for fractional optimal control problems. Nonlinear Dyn. 38, 323–337 (2004)

    Article  MATH  Google Scholar 

  3. Agrawal, O.M.P., Baleanu, D.: A Hamiltonian formulation and a direct numerical scheme for fractional optimal control problems. J. Vib. Cont. 13, 1269–1281 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  4. Almeida, R., Torres, D.F.M.: Necessary and sufficient conditions for the fractional calculus of variations with Caputo derivatives. Commun. Nonlinear Sci. Numer. Simul. 16, 1490–1500 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  5. Agrawal, O.M.P.: A formulation and numerical scheme for fractional optimal control problems. J. Vib. Cont. 14, 1291–1299 (2008)

    Article  MATH  Google Scholar 

  6. Almeida, R., Torres, D.F.M.: Calculus of variations with fractional derivatives and fractional integrals. Appl. Math. Lett. 22, 1816–1820 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  7. Agrawal, O.M.P.: Fractional variational calculus in terms of Riesz fractional derivatives. J. Phys. Math. Theo. 40, 6287–6303 (2007)

    Article  MATH  Google Scholar 

  8. Almeida, R., Torres, D.F.M.: Fractional variational calculus for nondifferentiable functions. Comput. Math. Appl. 61, 3097–3104 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  9. Agrawal, O.M.P.: Generalized variational problems and Euler-Lagrange equations. Comput. Math. Appl. 59, 1852–1864 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  10. Agrawal, O.M.P.: Fractional variational calculus and transversality conditions. J. Phys. Math. Gen. 39, 10375–10384 (2006)

    Article  MATH  Google Scholar 

  11. Agrawal, O.M.P.: Generalized Euler-Lagrange equations and transversality conditions for FVPs in terms of the Caputo Derivative. J. Vib. Cont. 13, 1217–1237 (2007)

    Article  MATH  Google Scholar 

  12. Malinowska, A.B., Torres, D.F.M.: Generalized natural boundary conditions for fractional variational problems in terms of the Caputo derivative. Comput. Math. Appl. 59, 3110–3116 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  13. Yousefi, S.A., Dehghan, M., Lotfi, A.: Generalized Euler-Lagrange equations for fractional variational problems with free boundary conditions. Comput. Math. Appl. 62, 987–995 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  14. Baleanu, D., Trujillo, J.J.: On exact solutions of class of fractional Euler-Lagrange equations. Nonlinear Dyn. 52, 331–335 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  15. Baleanu, D., Defterli, O., Agrawal, O.M.P.: A central difference numerical scheme for fractional optimal control problems. J. Vib. Cont. 15, 583–597 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  16. Tricaud, C., Chen, Y.Q.: An approximate method for numerically solving fractional order optimal control problems of general form. Comput. Math. Appl. 59, 1644–1655 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  17. Yousefi, S.A., Lotfi, A., Dehghan, M.: The use of a Legendre multiwavelet collocation method for solving the fractional optimal control problems. J. Vib. Cont. 17, 1–7 (2011)

    Article  MathSciNet  Google Scholar 

  18. Agrawal, O.M.P.: A general finite element formulation for fractional variational problems. J. Math. Anal. Appl. 337, 1–12 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  19. Agrawal, O.M.P., Mehedi Hasan, M., Tangpong, X.W.: A numerical scheme for a class of parametric problem of fractional variational calculus. J. Comput. Nonlinear Dyn. 7, 021005–1 (2012)

    Article  Google Scholar 

  20. Wang, D., Xiao, A.: Fractional variational integrators for fractional variational problems. Commun. Nonlinear Sci. Numer. Simul. 17, 602–610 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  21. Pooseh, S., Almeida, R., Torres, D.F.M.: Discrete direct methods in the fractional calculus of variations. Comput. Math. Appl. 66, 668–676 (2013)

    Article  MathSciNet  Google Scholar 

  22. Agrawal, O.M.P.: A quadratic numerical scheme for fractional optimal control problems. J. Dyn. Sys. Measur. Cont. 130, 011010-1-011010-6 (2008)

    Google Scholar 

  23. Lotfi, A., Yousefi, S.A.: A numerical technique for solving a class of fractional variational problems. J. Comput. Appl. Math. 237, 633–643 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  24. Lotfi, A., Dehghan, M., Yousefi, S.A.: A numerical technique for solving fractional optimal control problems. Comput. Math. Appl. 62, 1055–1067 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  25. Lotfi, A., Yousefi, S.A., Dehghan, M.: Numerical solution of a class of fractional optimal control problems via the Legendre orthonormal basis combined with the operational matrix and the Gauss quadrature rule. J. Comput. Appl. Math. 250, 143–160 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  26. Pooseh, S., Almeida, R., Torres, D.F.M.: Approximation of fractional integrals by means of derivatives. Comput. Math. appl. 64, 3090–3100 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  27. Malinowska, A.B., Torres, D.F.M.: Introduction to the Fractional Calculus of Variations. Imp. Coll. Press, London (2012)

    Book  MATH  Google Scholar 

  28. Balakrishnan, A.V.: On a new computing technique in optimal control. SIAM J. Cont. 6, 149–173 (1968)

    Article  MATH  Google Scholar 

  29. Frick, P.A.: An integral formulation of the \(\epsilon \)-problem and a new computational approach to control function optimization. J. Optim. Theo. Appl. 13, 553–581 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  30. Frick, P.A., Stech, D.J.: Epsilon-Ritz method for solving optimal control problems: usefull parallel solution method. J. Optim. Theo. Appl. 79, 31–58 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  31. Samko, S.G., Kilbas, A.A., Marichev, O.I.: Fractional Integrals and Derivatives. Theory and Applications. Gordon and Breach Sci. Publ, Langhorne (1993)

    MATH  Google Scholar 

  32. Podlubny, I.: Fractional Differential Equations. Academic Press, San Diego (1999)

    MATH  Google Scholar 

  33. Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. North-Holland Mathematics Studies, Amsterdam (2006)

    MATH  Google Scholar 

  34. Royden, H.L.: Real Analysis. Macmillan Publishing Company, U.S.A (1988)

    MATH  Google Scholar 

  35. Datta, K.B., Mohan, B.M.: Orthogonal Functions in Systems and Control. World Scientific, Singapore, New Jersey, London, Hong Kong (1995)

    Book  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Shahid Beheshti University, G.C, Tehran, Iran

    Ali Lotfi & Sohrab Ali Yousefi

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  1. Ali Lotfi
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  2. Sohrab Ali Yousefi
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Correspondence to Sohrab Ali Yousefi.

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Lotfi, A., Yousefi, S.A. Epsilon-Ritz Method for Solving a Class of Fractional Constrained Optimization Problems. J Optim Theory Appl 163, 884–899 (2014). https://doi.org/10.1007/s10957-013-0511-5

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  • DOI: https://doi.org/10.1007/s10957-013-0511-5

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