Generalized Euler—Lagrange Equations and Transversality Conditions for FVPs in terms of the Caputo Derivative

Abstract

This paper presents extensions to the traditional calculus of variations for systems containing Fractional Derivatives (FDs) defined in the Caputo sense. Specifically, two problems are considered, the simplest Fractional Variational Problem (FVP) and the FVP of Lagrange. Both specified and unspecified end conditions and end points are considered. Results of the first problem are extended to problems containing multiple FDs. For the second problem, we present a Lagrange type multiplier rule. For both problems, we develop generalized Euler—Lagrange type necessary conditions and the transversality conditions, which must be satisfied for the given functional to be extremum. The formulations show that one may require fractional boundary conditions even when the FVP has been formulated in terms of Caputo Fractional Derivatives (CFDs) only, and that the Riemann—Liouville Fractional Derivatives (RLFDs) automatically appear in the resulting equations. The relationships between the RLFDs and the CFDs are used to write the resulting differential equations purely in terms of the CFDs. Several examples are considered to demonstrate the application of the formulations. The formulations presented and the resulting equations are very similar to the formulations given for FVPs defined in terms of the RLFDs and to those that appear in the field of classical calculus of variations. The formulations presented are simple and can be extended to other problems in the field of fractional calculus of variations.

Keywords

Fractional derivative fractional calculus fractional calculus of variations fractional optimal control fractional variational problems.

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References

Riewe, F. , 1996, “Nonconservative Lagrangian and Hamiltonian mechanics ,” Physical Review E 53, 1890—1899.

Riewe, F. , 1997, “Mechanics with fractional derivatives,” Physical Review E 55, 3582—3592.

Agrawal, O.P. , 2002, “Formulation of Euler—Lagrange equations for fractional variational problems,” Journal of Mathematical Analysis and Applications 272, 368—379.

Klimek, M. , 2001, “Fractional sequential mechanics model with symmetric fractional derivatives,” Czechoslovak Journal of Physics 51, 1348—1354.

Klimek, M. , 2002, “Stationary conservation laws for fractional differential equations with variable coefficients,” Journal of Physics A: Mathematical and General 35, 6675—6693.

Dreisigmeyer, D.W. and Young, P.M. , 2003, “Nonconservative Lagrangian mechanics: A generalized function approach,” Journal of Physics A: Mathematical and General 36, 8297—8310.

Dreisigmeyer, D.W. and Young, P.M. , “ Extending Bauer's corollary to fractional derivatives,” Journal of Physics A: Mathematical and General 37, 117—121.

Baleanu, D. and Avkar, T. , 2004, “Lagrangians with linear velocities within Riemann-Liouville fractional derivatives,” Nuovo Cimento Della Societa Italiana Di Fisica B — General Physics Relativity Astronomy and Mathematical Physics and Methods 119, 73—79.

Baleanu, D. , 2004, “About metafluid dynamics,” Czechoslovak Journal of Physics 54 1165—1170.

10.

Muslih, S.I. and Baleanu, D. , 2005a, “Hamiltonian formulation of systems with linear velocities within Riemann— Liouville fractional derivatives,” Journal of Mathematical Analysis and Applications 304, 599—606.

11.

Muslih, S.I. and D. Baleanu , 2005b, “Formulation of Hamiltonian equations for fractional variational problems,” Czechoslovak Journal of Physics 55, 633—642.

12.

Agrawal, O.P. , 2004a, “A general formulation and solution scheme for fractional optimal control problems,” Nonlinear Dynamics 38, 323—337.

13.

Agrawal, O.P. , 2004b, “A General Scheme for Stochastic Analysis of Fractional Optimal Control Problems,” in Proceedings of the IFAC FDA04, Bordeaux, France, July, Paper No. 127.

14.

Tarasov, V.E. and Zaslavsky, G.M. , 2005, “Fractional Ginzburg—Landau equation for fractal media,” Physica A — Statistical Mechanics and Its Applications 354, 249—261.

15.

Podlubny, I. , 1999, Fractional Differential Equations, Academic Press, New York.

16.

Diethelm, K. , Ford, N.J. , Freed, A.D. , and Luchko, Y. , 2005, “Algorithms for the fractional calculus: A selection of numerical methods,” Computer Methods in Applied Mechanics and Engineering 194, 743—773.

17.

Samko, S.G. , Kilbas, A.A. , and Marichev, O.I. , 1993, Fractional Integrals and Derivatives — Theory and Applications, Gordon and Breach, Longhorne, PA.

18.

Gelfand, I.M. and Fomin, S.V. , 1963, Calculus of Variations, Prentice Hall , New Jersey.

19.

Agrawal, O.P. , Gregory, J. , and Spector, K.P. , 1997, “A Bliss-type multiplier rule for constrained variational problems with time delay,” Journal of Mathematical Analysis and Applications 210, 702—711.

20.

Agrawal, O.P. , 2006, “A formulation and a numerical scheme for fractional optimal control control problems,” in Proceedings of the IFAC/FDA06, Porto, Portugal, July 19—21.

21.

Agrawal, O.P. and Baleanu, D. , 2006, “A direct numerical scheme for fractional optimal control problems defined using RL derivatives,” in Proceedings of MME06, Ankara, Turkey, April 27—29.