# American Institute of Mathematical Sciences

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

## 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

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.

Citation: Ricardo Almeida. Optimality conditions for fractional variational problems with free terminal time. Discrete and 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. [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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##### 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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