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April  2011, 29(2): 417-437. doi: 10.3934/dcds.2011.29.417

## Necessary optimality conditions for fractional difference problems of the calculus of variations

 1 Department of Mathematics, ESTGV, Polytechnic Institute of Viseu, 3504-510 Viseu, Portugal 2 Faculty of Engineering and Natural Sciences, Lusophone University of Humanities and Technologies, 1749-024 Lisbon, Portugal 3 Department of Mathematics, University of Aveiro, 3810-193 Aveiro, Portugal

Received  May 2009 Revised  March 2010 Published  October 2010

We introduce a discrete-time fractional calculus of variations. First and second order necessary optimality conditions are established. Examples illustrating the use of the new Euler--Lagrange and Legendre type conditions are given. They show that the solutions of the fractional problems coincide with the solutions of the corresponding non-fractional variational problems when the order of the discrete derivatives is an integer value.
Citation: Nuno R. O. Bastos, Rui A. C. Ferreira, Delfim F. M. Torres. Necessary optimality conditions for fractional difference problems of the calculus of variations. Discrete and Continuous Dynamical Systems, 2011, 29 (2) : 417-437. doi: 10.3934/dcds.2011.29.417
##### References:
 [1] O. P. Agrawal, Formulation of Euler-Lagrange equations for fractional variational problems, J. Math. Anal. Appl., 272 (2002), 368-379. doi: doi:10.1016/S0022-247X(02)00180-4. [2] O. P. Agrawal, A general formulation and solution scheme for fractional optimal control problems, Nonlinear Dynam., 38 (2004), 323-337. doi: doi:10.1007/s11071-004-3764-6. [3] O. P. Agrawal, Fractional variational calculus and the transversality conditions, J. Phys. A, 39 (2006), 10375-10384. doi: doi:10.1088/0305-4470/39/33/008. [4] O. P. Agrawal, A general finite element formulation for fractional variational problems, J. Math. Anal. Appl., 337 (2008), 1-12. doi: doi:10.1016/j.jmaa.2007.03.105. [5] O. P. Agrawal and D. Baleanu, A Hamiltonian formulation and a direct numerical scheme for fractional optimal control problems, J. Vib. Control, 13 (2007), 1269-1281. doi: doi:10.1177/1077546307077467. [6] E. Akin, Cauchy functions for dynamic equations on a measure chain, J. Math. Anal. Appl., 267 (2002), 97-115. doi: doi:10.1006/jmaa.2001.7753. [7] R. Almeida, A. B. Malinowska and D. F. M. Torres, A fractional calculus of variations for multiple integrals with application to vibrating string, J. Math. Phys., 51 (2010), 033503. doi: doi:10.1063/1.3319559. [8] R. Almeida and D. F. M. Torres, Calculus of variations with fractional derivatives and fractional integrals, Appl. Math. Lett., 22 (2009), 1816-1820. doi: doi:10.1016/j.aml.2009.07.002. [9] F. M. Atici and P. W. Eloe, A transform method in discrete fractional calculus, Int. J. Difference Equ., 2 (2007), 165-176. doi: doi:10.1090/S0002-9939-08-09626-3. [10] F. M. Atici and P. W. Eloe, Initial value problems in discrete fractional calculus, Proc. Amer. Math. Soc., 137 (2009), 981-989. doi: doi:10.1177/1077546308088565. [11] D. Baleanu, O. Defterli and O. P. Agrawal, A central difference numerical scheme for fractional optimal control problems, J. Vib. Control, 15 (2009), 583-597. [12] M. Bohner, Calculus of variations on time scales, Dynam. Systems Appl., 13 (2004), 339-349. [13] R. A. El-Nabulsi and D. F. M. Torres, Necessary optimality conditions for fractional action-like integrals of variational calculus with Riemann-Liouville derivatives of order $(\alpha,\beta)$, Math. Meth. Appl. Sci., 30 (2007), 1931-1939. doi: doi:10.1002/mma.879. [14] R. A. C. Ferreira and D. F. M. Torres, Higher-order calculus of variations on time scales, in "Mathematical Control Theory and Finance" (eds. A. Sarychev, A. Shiryaev, M. Guerra and M. R. Grossinho), Springer, Berlin, (2008), 149-159. doi: doi:10.1007/978-3-540-69532-5_9. [15] 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., 334 (2007), 834-846. doi: doi:10.1016/j.jmaa.2007.01.013. [16] G. S. F. Frederico and D. F. M. Torres, Fractional conservation laws in optimal control theory, Nonlinear Dynam., 53 (2008), 215-222. doi: doi:10.1007/s11071-007-9309-z. [17] R. Hilscher and V. Zeidan, Nonnegativity of a discrete quadratic functional in terms of the (strengthened) Legendre and Jacobi conditions, Comput. Math. Appl., 45 (2003), 1369-1383. doi: doi:10.1016/S0898-1221(03)00109-3. [18] R. Hilscher and V. Zeidan, Nonnegativity and positivity of quadratic functionals in discrete calculus of variations: Survey, J. Difference Equ. Appl., 11 (2005), 857-875. doi: doi:10.1080/10236190500137454. [19] W. G. Kelley and A. C. Peterson, "Difference Equations," Academic Press, Boston, MA, 1991. [20] A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, "Theory and Applications of Fractional Differential Equations," Elsevier, Amsterdam, 2006. [21] A. B. Malinowska and D. F. M. Torres, Generalized natural boundary conditions for fractional variational problems in terms of the Caputo derivative, Comput. Math. Appl., 59 (2010), 3110-3116. doi: doi:10.1016/j.camwa.2010.02.032. [22] K. S. Miller and B. Ross, Fractional difference calculus, in "Univalent Functions, Fractional Calculus, and Their Applications" (eds. H. M. Srivastava and S. Owa), (Kōriyama, 1988), Horwood, Chichester, (1989), 139-152. [23] K. S. Miller and B. Ross, "An Introduction to the Fractional Calculus and Fractional Differential Equations," John Wiley and Sons, Inc., New York, 1993. [24] M. D. Ortigueira, Fractional central differences and derivatives, J. Vib. Control, 14 (2008), 1255-1266. doi: doi:10.1177/1077546307087453. [25] M. D. Ortigueira and A. G. Batista, A fractional linear system view of the fractional Brownian motion, Nonlinear Dynam., 38 (2004), 295-303. doi: doi:10.1007/s11071-004-3762-8. [26] F. Riewe, Nonconservative Lagrangian and Hamiltonian mechanics, Phys. Rev. E, 53 (1996), 1890-1899. doi: doi:10.1103/PhysRevE.53.1890. [27] S. G. Samko, A. A. Kilbas and O. I. Marichev, "Fractional Integrals and Derivatives," Translated from the 1987 Russian original, Gordon and Breach, Yverdon, 1993. [28] M. F. Silva, J. A. Tenreiro Machado and R. S. Barbosa, Using fractional derivatives in joint control of hexapod robots, J. Vib. Control, 14 (2008), 1473-1485. doi: doi:10.1177/1077546307087436.

show all references

##### References:
 [1] O. P. Agrawal, Formulation of Euler-Lagrange equations for fractional variational problems, J. Math. Anal. Appl., 272 (2002), 368-379. doi: doi:10.1016/S0022-247X(02)00180-4. [2] O. P. Agrawal, A general formulation and solution scheme for fractional optimal control problems, Nonlinear Dynam., 38 (2004), 323-337. doi: doi:10.1007/s11071-004-3764-6. [3] O. P. Agrawal, Fractional variational calculus and the transversality conditions, J. Phys. A, 39 (2006), 10375-10384. doi: doi:10.1088/0305-4470/39/33/008. [4] O. P. Agrawal, A general finite element formulation for fractional variational problems, J. Math. Anal. Appl., 337 (2008), 1-12. doi: doi:10.1016/j.jmaa.2007.03.105. [5] O. P. Agrawal and D. Baleanu, A Hamiltonian formulation and a direct numerical scheme for fractional optimal control problems, J. Vib. Control, 13 (2007), 1269-1281. doi: doi:10.1177/1077546307077467. [6] E. Akin, Cauchy functions for dynamic equations on a measure chain, J. Math. Anal. Appl., 267 (2002), 97-115. doi: doi:10.1006/jmaa.2001.7753. [7] R. Almeida, A. B. Malinowska and D. F. M. Torres, A fractional calculus of variations for multiple integrals with application to vibrating string, J. Math. Phys., 51 (2010), 033503. doi: doi:10.1063/1.3319559. [8] R. Almeida and D. F. M. Torres, Calculus of variations with fractional derivatives and fractional integrals, Appl. Math. Lett., 22 (2009), 1816-1820. doi: doi:10.1016/j.aml.2009.07.002. [9] F. M. Atici and P. W. Eloe, A transform method in discrete fractional calculus, Int. J. Difference Equ., 2 (2007), 165-176. doi: doi:10.1090/S0002-9939-08-09626-3. [10] F. M. Atici and P. W. Eloe, Initial value problems in discrete fractional calculus, Proc. Amer. Math. Soc., 137 (2009), 981-989. doi: doi:10.1177/1077546308088565. [11] D. Baleanu, O. Defterli and O. P. Agrawal, A central difference numerical scheme for fractional optimal control problems, J. Vib. Control, 15 (2009), 583-597. [12] M. Bohner, Calculus of variations on time scales, Dynam. Systems Appl., 13 (2004), 339-349. [13] R. A. El-Nabulsi and D. F. M. Torres, Necessary optimality conditions for fractional action-like integrals of variational calculus with Riemann-Liouville derivatives of order $(\alpha,\beta)$, Math. Meth. Appl. Sci., 30 (2007), 1931-1939. doi: doi:10.1002/mma.879. [14] R. A. C. Ferreira and D. F. M. Torres, Higher-order calculus of variations on time scales, in "Mathematical Control Theory and Finance" (eds. A. Sarychev, A. Shiryaev, M. Guerra and M. R. Grossinho), Springer, Berlin, (2008), 149-159. doi: doi:10.1007/978-3-540-69532-5_9. [15] 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., 334 (2007), 834-846. doi: doi:10.1016/j.jmaa.2007.01.013. [16] G. S. F. Frederico and D. F. M. Torres, Fractional conservation laws in optimal control theory, Nonlinear Dynam., 53 (2008), 215-222. doi: doi:10.1007/s11071-007-9309-z. [17] R. Hilscher and V. Zeidan, Nonnegativity of a discrete quadratic functional in terms of the (strengthened) Legendre and Jacobi conditions, Comput. Math. Appl., 45 (2003), 1369-1383. doi: doi:10.1016/S0898-1221(03)00109-3. [18] R. Hilscher and V. Zeidan, Nonnegativity and positivity of quadratic functionals in discrete calculus of variations: Survey, J. Difference Equ. Appl., 11 (2005), 857-875. doi: doi:10.1080/10236190500137454. [19] W. G. Kelley and A. C. Peterson, "Difference Equations," Academic Press, Boston, MA, 1991. [20] A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, "Theory and Applications of Fractional Differential Equations," Elsevier, Amsterdam, 2006. [21] A. B. Malinowska and D. F. M. Torres, Generalized natural boundary conditions for fractional variational problems in terms of the Caputo derivative, Comput. Math. Appl., 59 (2010), 3110-3116. doi: doi:10.1016/j.camwa.2010.02.032. [22] K. S. Miller and B. Ross, Fractional difference calculus, in "Univalent Functions, Fractional Calculus, and Their Applications" (eds. H. M. Srivastava and S. Owa), (Kōriyama, 1988), Horwood, Chichester, (1989), 139-152. [23] K. S. Miller and B. Ross, "An Introduction to the Fractional Calculus and Fractional Differential Equations," John Wiley and Sons, Inc., New York, 1993. [24] M. D. Ortigueira, Fractional central differences and derivatives, J. Vib. Control, 14 (2008), 1255-1266. doi: doi:10.1177/1077546307087453. [25] M. D. Ortigueira and A. G. Batista, A fractional linear system view of the fractional Brownian motion, Nonlinear Dynam., 38 (2004), 295-303. doi: doi:10.1007/s11071-004-3762-8. [26] F. Riewe, Nonconservative Lagrangian and Hamiltonian mechanics, Phys. Rev. E, 53 (1996), 1890-1899. doi: doi:10.1103/PhysRevE.53.1890. [27] S. G. Samko, A. A. Kilbas and O. I. Marichev, "Fractional Integrals and Derivatives," Translated from the 1987 Russian original, Gordon and Breach, Yverdon, 1993. [28] M. F. Silva, J. A. Tenreiro Machado and R. S. Barbosa, Using fractional derivatives in joint control of hexapod robots, J. Vib. Control, 14 (2008), 1473-1485. doi: doi:10.1177/1077546307087436.
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