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On a generalization of the impulsive control concept: Controlling system jumps
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 |
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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