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February  2018, 11(1): 1-19. doi: 10.3934/dcdss.2018001

Optimality conditions for fractional variational problems with free terminal time

Ricardo Almeida

Center for Research and Development in Mathematics and Applications (CIDMA), Department of Mathematics, University of Aveiro, 3810-193 Aveiro, Portugal

Received  June 2016 Revised  January 2017 Published  January 2018

Fund Project: Work supported by Portuguese funds through the CIDMA -Center for Research and Development in Mathematics and Applications, and the Portuguese Foundation for Science and Technology (FCT-Fundação para a Ciência e a Tecnologia), within project UID/MAT/04106/2013. The author is grateful to two Reviewers, for several pertinent remarks, which improved the final version of the manuscript.

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.

Keywords: Fractional calculus, Euler-Lagrange equation, Legendre condition, isoperimetric problem.
Mathematics Subject Classification: Primary: 26A33, 49K05; Secondary: 34A08.
Citation: Ricardo Almeida. Optimality conditions for fractional variational problems with free terminal time. Discrete & Continuous Dynamical Systems - S, 2018, 11 (1) : 1-19. doi: 10.3934/dcdss.2018001
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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

D. Baleanu, New applications of fractional variational principles, Rep. Math. Phys., 61 (2008), 199-206.  doi: 10.1016/S0034-4877(08)80007-9.  Google Scholar

[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.  Google Scholar

[15]

B. van Brunt, The Calculus of Variations Universitext, Springer, New York, 2004. doi: 10.1007/b97436.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  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: 10.1016/S0022-247X(02)00180-4.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

D. Baleanu, New applications of fractional variational principles, Rep. Math. Phys., 61 (2008), 199-206.  doi: 10.1016/S0034-4877(08)80007-9.  Google Scholar

[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.  Google Scholar

[15]

B. van Brunt, The Calculus of Variations Universitext, Springer, New York, 2004. doi: 10.1007/b97436.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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