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Preface
Optimality conditions for fractional variational problems with free terminal time
Center for Research and Development in Mathematics and Applications (CIDMA), Department of Mathematics, University of Aveiro, 3810-193 Aveiro, Portugal |
This paper provides necessary and sufficient conditions of optimality for variational problems that deal with a fractional derivative with respect to another function. Fractional Euler-Lagrange equations are established for the fundamental problem and when in presence of an integral constraint. A Legendre condition, which is a second-order necessary condition, is also obtained. Other cases, such as the infinite horizon problem, the problem with delays in the Lagrangian, and the problem with high-order derivatives, are considered. Finally, a necessary condition for the optimal fractional order to satisfy is proved.
References:
[1] |
O. P. Agrawal,
Formulation of Euler-Lagrange equations for fractional variational problems, J. Math. Anal. Appl., 272 (2002), 368-379.
doi: 10.1016/S0022-247X(02)00180-4. |
[2] |
O. P. Agrawal,
Fractional variational calculus and the transversality conditions, J. Phys. A, 39 (2006), 10375-10384.
doi: 10.1088/0305-4470/39/33/008. |
[3] |
O. P. Agrawal,
Fractional variational calculus in terms of Riesz fractional derivatives, J. Phys. A, 40 (2007), 6287-6303.
doi: 10.1088/1751-8113/40/24/003. |
[4] |
R. Almeida,
Fractional variational problems with the Riesz-Caputo derivative, Appl. Math. Lett., 25 (2012), 142-148.
doi: 10.1016/j.aml.2011.08.003. |
[5] |
R. Almeida,
Fractional variational problems depending on indefinite integrals and with delay, Bull. Malays. Math. Sci. Soc., 39 (2016), 1515-1528.
doi: 10.1007/s40840-015-0248-4. |
[6] |
R. Almeida,
A Caputo fractional derivative of a function with respect to another function, Commun. Nonlinear Sci. Numer. Simul., 44 (2017), 460-481.
doi: 10.1016/j.cnsns.2016.09.006. |
[7] |
R. Almeida, R. A. C. Ferreira and D. F. M. Torres,
Isoperimetric problems of the calculus of variations with fractional derivatives, Acta Math. Sci. Ser. B Engl. Ed., 32 (2012), 619-630.
doi: 10.1016/S0252-9602(12)60043-5. |
[8] |
R. Almeida and A. B. Malinowska,
Generalized transversality conditions in fractional calculus of variations, Commun. Nonlinear Sci. Numer. Simul., 18 (2013), 443-452.
doi: 10.1016/j.cnsns.2012.07.009. |
[9] |
R. Almeida, S. Pooseh and D. F. M. Torres,
Computational Methods in the Fractional Calculus of Variations Imp. Coll. Press, London, 2015.
doi: 10.1142/p991. |
[10] |
T. M. Atanacković, S. Konjik and S. Pilipović, Variational problems with fractional derivatives: Euler-Lagrange equations J. Phys. A 41 (2008), 095201, 12 pp.
doi: 10.1088/1751-8113/41/9/095201. |
[11] |
D. Baleanu and S. I. Muslih,
Lagrangian formulation of classical fields within Riemann-Liouville fractional derivatives, Phys. Scripta, 72 (2005), 119-121.
doi: 10.1238/Physica.Regular.072a00119. |
[12] |
D. Baleanu, T. Maaraba and F. Jarad, Fractional variational principles with delay J. Phys. A 41(2008), 315403, 8pp.
doi: 10.1088/1751-8113/41/31/315403. |
[13] |
D. Baleanu,
New applications of fractional variational principles, Rep. Math. Phys., 61 (2008), 199-206.
doi: 10.1016/S0034-4877(08)80007-9. |
[14] |
W. A. Brock,
On existence of weakly maximal programmes in a multi-sector economy, Rev. Econom. Stud., 37 (1970), 275-280.
doi: 10.2307/2296419. |
[15] |
B. van Brunt,
The Calculus of Variations Universitext, Springer, New York, 2004.
doi: 10.1007/b97436. |
[16] |
M. A. E. Herzallah and D. Baleanu,
Fractional-order Euler-Lagrange equations and formulation of Hamiltonian equations, Nonlinear Dynam., 58 (2009), 385-391.
doi: 10.1007/s11071-009-9486-z. |
[17] |
F. Jarad, T. Abdeljawad and D. Baleanu,
Fractional variational principles with delay within Caputo derivatives, Rep. Math. Phys., 65 (2010), 17-28.
doi: 10.1016/S0034-4877(10)00010-8. |
[18] |
M. J. Lazo and D. F. M. Torres,
The Legendre condition of the fractional calculus of variations, Optimization, 63 (2014), 1157-1165.
doi: 10.1080/02331934.2013.877908. |
[19] |
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: 10.1016/j.camwa.2010.02.032. |
[20] |
A. B. Malinowska and D. F. M. Torres,
Introduction to the Fractional Calculus of Variations Imp. Coll. Press, London, 2012.
doi: 10.1142/p871. |
[21] |
T. Odziehjewicz,
Generalized fractional isoperimetric problem of several variables, Discrete Contin. Dyn. Syst. Ser. B, 19 (2014), 2617-2629.
doi: 10.3934/dcdsb.2014.19.2617. |
[22] |
T. Odzijewicz, A. B. Malinowska and D. F. M. Torres,
Fractional variational calculus with classical and combined Caputo derivatives, Nonlinear Anal., 75 (2012), 1507-1515.
doi: 10.1016/j.na.2011.01.010. |
[23] |
J.-P. Richard,
Time-delay systems: An overview of some recent advances and open problems, Automatica J. IFAC, 39 (2003), 1667-1694.
doi: 10.1016/S0005-1098(03)00167-5. |
[24] |
D. Salamon,
On controllability and observability of time delay systems, IEEE Trans. Automat. Control, 29 (1984), 432-439.
doi: 10.1109/TAC.1984.1103560. |
[25] |
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. |
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: 10.1016/S0022-247X(02)00180-4. |
[2] |
O. P. Agrawal,
Fractional variational calculus and the transversality conditions, J. Phys. A, 39 (2006), 10375-10384.
doi: 10.1088/0305-4470/39/33/008. |
[3] |
O. P. Agrawal,
Fractional variational calculus in terms of Riesz fractional derivatives, J. Phys. A, 40 (2007), 6287-6303.
doi: 10.1088/1751-8113/40/24/003. |
[4] |
R. Almeida,
Fractional variational problems with the Riesz-Caputo derivative, Appl. Math. Lett., 25 (2012), 142-148.
doi: 10.1016/j.aml.2011.08.003. |
[5] |
R. Almeida,
Fractional variational problems depending on indefinite integrals and with delay, Bull. Malays. Math. Sci. Soc., 39 (2016), 1515-1528.
doi: 10.1007/s40840-015-0248-4. |
[6] |
R. Almeida,
A Caputo fractional derivative of a function with respect to another function, Commun. Nonlinear Sci. Numer. Simul., 44 (2017), 460-481.
doi: 10.1016/j.cnsns.2016.09.006. |
[7] |
R. Almeida, R. A. C. Ferreira and D. F. M. Torres,
Isoperimetric problems of the calculus of variations with fractional derivatives, Acta Math. Sci. Ser. B Engl. Ed., 32 (2012), 619-630.
doi: 10.1016/S0252-9602(12)60043-5. |
[8] |
R. Almeida and A. B. Malinowska,
Generalized transversality conditions in fractional calculus of variations, Commun. Nonlinear Sci. Numer. Simul., 18 (2013), 443-452.
doi: 10.1016/j.cnsns.2012.07.009. |
[9] |
R. Almeida, S. Pooseh and D. F. M. Torres,
Computational Methods in the Fractional Calculus of Variations Imp. Coll. Press, London, 2015.
doi: 10.1142/p991. |
[10] |
T. M. Atanacković, S. Konjik and S. Pilipović, Variational problems with fractional derivatives: Euler-Lagrange equations J. Phys. A 41 (2008), 095201, 12 pp.
doi: 10.1088/1751-8113/41/9/095201. |
[11] |
D. Baleanu and S. I. Muslih,
Lagrangian formulation of classical fields within Riemann-Liouville fractional derivatives, Phys. Scripta, 72 (2005), 119-121.
doi: 10.1238/Physica.Regular.072a00119. |
[12] |
D. Baleanu, T. Maaraba and F. Jarad, Fractional variational principles with delay J. Phys. A 41(2008), 315403, 8pp.
doi: 10.1088/1751-8113/41/31/315403. |
[13] |
D. Baleanu,
New applications of fractional variational principles, Rep. Math. Phys., 61 (2008), 199-206.
doi: 10.1016/S0034-4877(08)80007-9. |
[14] |
W. A. Brock,
On existence of weakly maximal programmes in a multi-sector economy, Rev. Econom. Stud., 37 (1970), 275-280.
doi: 10.2307/2296419. |
[15] |
B. van Brunt,
The Calculus of Variations Universitext, Springer, New York, 2004.
doi: 10.1007/b97436. |
[16] |
M. A. E. Herzallah and D. Baleanu,
Fractional-order Euler-Lagrange equations and formulation of Hamiltonian equations, Nonlinear Dynam., 58 (2009), 385-391.
doi: 10.1007/s11071-009-9486-z. |
[17] |
F. Jarad, T. Abdeljawad and D. Baleanu,
Fractional variational principles with delay within Caputo derivatives, Rep. Math. Phys., 65 (2010), 17-28.
doi: 10.1016/S0034-4877(10)00010-8. |
[18] |
M. J. Lazo and D. F. M. Torres,
The Legendre condition of the fractional calculus of variations, Optimization, 63 (2014), 1157-1165.
doi: 10.1080/02331934.2013.877908. |
[19] |
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: 10.1016/j.camwa.2010.02.032. |
[20] |
A. B. Malinowska and D. F. M. Torres,
Introduction to the Fractional Calculus of Variations Imp. Coll. Press, London, 2012.
doi: 10.1142/p871. |
[21] |
T. Odziehjewicz,
Generalized fractional isoperimetric problem of several variables, Discrete Contin. Dyn. Syst. Ser. B, 19 (2014), 2617-2629.
doi: 10.3934/dcdsb.2014.19.2617. |
[22] |
T. Odzijewicz, A. B. Malinowska and D. F. M. Torres,
Fractional variational calculus with classical and combined Caputo derivatives, Nonlinear Anal., 75 (2012), 1507-1515.
doi: 10.1016/j.na.2011.01.010. |
[23] |
J.-P. Richard,
Time-delay systems: An overview of some recent advances and open problems, Automatica J. IFAC, 39 (2003), 1667-1694.
doi: 10.1016/S0005-1098(03)00167-5. |
[24] |
D. Salamon,
On controllability and observability of time delay systems, IEEE Trans. Automat. Control, 29 (1984), 432-439.
doi: 10.1109/TAC.1984.1103560. |
[25] |
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. |

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