Discrete-time fractional variational problems

Signal Processing

Volumn 91, Issue 3, 2011, Pages 513-524

  Bastos, Nuno R O ^a   Ferreira, Rui A C ^b   Torres, Delfim F M ^c  

^a POLYTECHNIC INSTITUTE OF VISEU   (Portugal)

^b UNIVERSIDADE LUSÓFONA DE HUMANIDADES E TECNOLOGIAS   (Portugal)

^c UNIVERSITY OF AVEIRO   (Portugal)

Author keywords

Calculus of variations; Euler Lagrange equation; Fractional difference calculus; Fractional summation by parts; Legendre necessary condition; Natural boundary conditions; Time scale hZ

Indexed keywords

CALCULUS OF VARIATIONS; EULER-LAGRANGE EQUATIONS; FRACTIONAL DIFFERENCE CALCULUS; FRACTIONAL SUMMATION BY PARTS; LEGENDRE; NATURAL BOUNDARY CONDITION; TIME-SCALES;

BOUNDARY CONDITIONS; CALCULATIONS; EQUATIONS OF MOTION; LAGRANGE MULTIPLIERS; SOLAR CELL ARRAYS; TIME MEASUREMENT;

DIFFERENCE EQUATIONS;

EID: 78049333168     PISSN: 01651684     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.sigpro.2010.05.001     Document Type: Article

References (50)

1. O.P. Agrawal Formulation of Euler-Lagrange equations for fractional variational problems J. Math. Anal. Appl. 272 1 2002 368 379
2. O.P. Agrawal A general finite element formulation for fractional variational problems J. Math. Anal. Appl. 337 1 2008 1 12
3. O.P. Agrawal, and D. Baleanu A Hamiltonian formulation and a direct numerical scheme for fractional optimal control problems J. Vib. Control 13 910 2007 1269 1281
4. 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 3 2010 033503 12pp
5. R. Almeida, and D.F.M. Torres Hlderian variational problems subject to integral constraints J. Math. Anal. Appl. 359 2 2009 674 681
6. R. Almeida, and D.F.M. Torres Isoperimetric problems on time scales with nabla derivatives J. Vib. Control 15 6 2009 951 958
7. R. Almeida, and D.F.M. Torres Calculus of variations with fractional derivatives and fractional integrals Appl. Math. Lett. 22 12 2009 1816 1820
8. F.M. Atici, and P.W. Eloe A transform method in discrete fractional calculus Int. J. Difference Equ. 2 2 2007 165 176
9. F.M. Atici, and P.W. Eloe Initial value problems in discrete fractional calculus Proc. Amer. Math. Soc. 137 3 2009 981 989
10. D. Baleanu New applications of fractional variational principles Rep. Math. Phys. 61 2 2008 199 206
11. D. Baleanu, O. Defterli, and O.P. Agrawal A central difference numerical scheme for fractional optimal control problems J. Vib. Control 15 4 2009 583 597
12. D. Baleanu, and F. Jarad Discrete variational principles for higher-order Lagrangians Nuovo Cimento Soc. Ital. Fis. B 120 9 2005 931 938
13. D. Baleanu, F. Jarad, Difference discrete variational principles, in: Mathematical Analysis and Applications, American Institute of Physics, Melville, NY, 2006 pp. 2029.
14. D. Baleanu, T. Maaraba, and F. Jarad Fractional variational principles with delay J. Phys. A 41 31 2008 315403 8pp
15. D. Baleanu, and S.I. Muslih Nonconservative systems within fractional generalized derivatives J. Vib. Control 14 910 2008 1301 1311
16. D. Baleanu, S.I. Muslih, and E.M. Rabei On fractional Euler-Lagrange and Hamilton equations and the fractional generalization of total time derivative Nonlinear Dynam. 53 12 2008 67 74
17. Z. Bartosiewicz, and D.F.M. Torres Noether's theorem on time scales J. Math. Anal. Appl. 342 2 2008 1220 1226
18. M. Bohner Calculus of variations on time scales Dynam. Syst. Appl. 13 2004 339 349
19. Bohner M., Ferreira R.A.C., and Torres D.F.M. Integral inequalities and their applications to the calculus of variations on time scales Math. Inequal. Appl. 13 3 2010 511 522
20. M. Bohner, and A. Peterson Dynamic Equations on Time Scales 2001 Birkhuser Boston, MA
21. J. Cresson, G.S.F. Frederico, and D.F.M. Torres Constants of motion for non-differentiable quantum variational problems Topol. Methods Nonlinear Anal. 33 2 2009 217 231
22. R.A. El-Nabulsi, and D.F.M. Torres Necessary optimality conditions for fractional action-like integrals of variational calculus with RiemannLiouville derivatives of order (α, β ) Math. Methods Appl. Sci. 30 15 2007 1931 1939
23. R.A. El-Nabulsi, and D.F.M. Torres Fractional action like variational problems J. Math. Phys. 49 5 2008 053521 7pp
24. R.A.C. Ferreira, D.F.M. Torres, Higher-order calculus of variations on time scales, in: Mathematical Control Theory and Finance, Springer, Berlin, 2008, pp. 149159.
25. 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 2 2007 834 846
26. G.S.F. Frederico, and D.F.M. Torres Fractional conservation laws in optimal control theory Nonlinear Dynam. 53 3 2008 215 222
27. G.S.F. Frederico, D.F.M. Torres, Fractional Noether's theorem in the RieszCaputo sense, Appl. Math. Comput. (2010), in press. DOI: 10.1016/j.amc.2010.01.100.
28. rd solutions with variable endpoints J. Math. Anal. Appl. 289 1 2004 143 166
29. W.G. Kelley, and A.C. Peterson Difference Equations 1991 Academic Press Boston, MA
30. A.A. Kilbas, H.M. Srivastava, and J.J. Trujillo Theory and Applications of Fractional Differential Equations 2006 Elsevier Amsterdam
31. F. Jarad, and D. Baleanu Discrete variational principles for Lagrangians linear in velocities Rep. Math. Phys. 59 1 2007 33 43
32. F. Jarad, D. Baleanu, and T. Maraaba Hamiltonian formulation of singular Lagrangians on time scales Chin. Phys. Lett. 25 5 2008 1720 1723
33. A.B. Malinowska, and D.F.M. Torres Strong minimizers of the calculus of variations on time scales and the Weierstrass condition Proc. Est. Acad. Sci. 58 4 2009 205 212
34. A.B. Malinowska, D.F.M. Torres, Natural boundary conditions in the calculus of variations, Math. Methods Appl. Sci. (2010), in press. DOI: 10.1002/mma.1289.
35. 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 9 2010 3110 3116
36. A.B. Malinowska, D.F.M. Torres, Leitmann's direct method of optimization for absolute extrema of certain problems of the calculus of variations on time scales, Appl. Math. Comput. (2010), in press. DOI: 10.1016/j.amc.2010.01.015.
37. T. Maraaba, F. Jarad, D. Baleanu, Variational optimal-control problems with delayed arguments on time scales, Adv. Difference Equ. (2009), Art. ID 840386, 15pp.
38. N. Martins, and D.F.M. Torres Calculus of variations on time scales with nabla derivatives Nonlinear Anal. 71 12 2009 e763 e773
39. K.S. Miller, B. Ross, Fractional difference calculus, in: Univalent Functions, Fractional Calculus, and their Applications (Kriyama, 1988), Horwood, Chichester, 1989, pp. 139152.
40. K.S. Miller, and B. Ross An Introduction to the Fractional Calculus and Fractional Differential Equations 1993 Wiley New York
41. S.I. Muslih, and D. Baleanu Fractional Euler-Lagrange equations of motion in fractional space J. Vib. Control 13 910 2007 1209 1216 (Pubitemid 47482808)
42. M.D. Ortigueira Fractional central differences and derivatives J. Vib. Control 14 910 2008 1255 1266
43. I. Podlubny Fractional Differential Equations 1999 Academic Press San Diego, CA
44. E.M. Rabei, K.I. Nawafleh, R.S. Hijjawi, S.I. Muslih, and D. Baleanu The Hamilton formalism with fractional derivatives J. Math. Anal. Appl. 327 2 2007 891 897
45. E.M. Rabei, D.M. Tarawneh, S.I. Muslih, and D. Baleanu Heisenberg's equations of motion with fractional derivatives J. Vib. Control 13 910 2007 1239 1247 (Pubitemid 47482810)
46. B. Ross, S.G. Samko, and E.R. Love Functions that have no first order derivative might have fractional derivatives of all orders less than one Real Anal. Exchange 20 1994 140 157
47. S.G. Samko, A.A. Kilbas, O.I. Marichev, Fractional Integrals and Derivatives, Translated from the 1987 Russian original, Gordon and Breach, Yverdon, 1993.
48. M.F. Silva, J.A. Tenreiro Machado, and R.S. Barbosa Using fractional derivatives in joint control of hexapod robots J. Vib. Control 14 910 2008 1473 1485
49. D.F.M. Torres, and G. Leitmann Contrasting two transformation-based methods for obtaining absolute extrema J. Optim. Theory Appl. 137 1 2008 53 59
50. B. van Brunt The Calculus of Variations 2004 Springer New York