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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

Nuno R. O. Bastos 1, , Rui A. C. Ferreira 2, and  Delfim F. M. Torres 3,

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.
Keywords: Legendre necessary condition., Fractional difference calculus, fractional summation by parts, Euler-Lagrange equation, calculus of variations.
Mathematics Subject Classification: 49K05, 39A12, 26A3.
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 & 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

M. Bohner, Calculus of variations on time scales, Dynam. Systems Appl., 13 (2004), 339-349.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

W. G. Kelley and A. C. Peterson, "Difference Equations," Academic Press, Boston, MA, 1991.  Google Scholar

[20]

A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, "Theory and Applications of Fractional Differential Equations," Elsevier, Amsterdam, 2006.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[24]

M. D. Ortigueira, Fractional central differences and derivatives, J. Vib. Control, 14 (2008), 1255-1266. doi: doi:10.1177/1077546307087453.  Google Scholar

[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.  Google Scholar

[26]

F. Riewe, Nonconservative Lagrangian and Hamiltonian mechanics, Phys. Rev. E, 53 (1996), 1890-1899. doi: doi:10.1103/PhysRevE.53.1890.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

M. Bohner, Calculus of variations on time scales, Dynam. Systems Appl., 13 (2004), 339-349.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

W. G. Kelley and A. C. Peterson, "Difference Equations," Academic Press, Boston, MA, 1991.  Google Scholar

[20]

A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, "Theory and Applications of Fractional Differential Equations," Elsevier, Amsterdam, 2006.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[24]

M. D. Ortigueira, Fractional central differences and derivatives, J. Vib. Control, 14 (2008), 1255-1266. doi: doi:10.1177/1077546307087453.  Google Scholar

[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.  Google Scholar

[26]

F. Riewe, Nonconservative Lagrangian and Hamiltonian mechanics, Phys. Rev. E, 53 (1996), 1890-1899. doi: doi:10.1103/PhysRevE.53.1890.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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