## Fractional variational calculus for nondifferentiable functions.(English)Zbl 1222.49026

Summary: We prove necessary optimality conditions, in the class of continuous functions, for variational problems defined with Jumarie’s modified Riemann-Liouville derivative. The fractional basic problem of the calculus of variations with free boundary conditions is considered, as well as problems with isoperimetric and holonomic constraints.

### MSC:

 49K05 Optimality conditions for free problems in one independent variable 26A33 Fractional derivatives and integrals
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### References:

 [1] Miller, K.S.; Ross, B., An introduction to the fractional calculus and fractional differential equations, (1993), Wiley New York · Zbl 0789.26002 [2] Podlubny, I., Fractional differential equations, (1999), Academic Press San Diego, CA · Zbl 0918.34010 [3] Samko, S.G.; Kilbas, A.A.; Marichev, O.I., (), Translated from the 1987 Russian original · Zbl 0818.26003 [4] Ross, B.; Samko, S.G.; Love, E.R., Functions that have no first order derivative might have fractional derivatives of all orders less than one, Real anal. exchange, 20, 1, 140-157, (1994-1995) · Zbl 0820.26002 [5] Jumarie, G., On the representation of fractional Brownian motion as an integral with respect to $$(\operatorname{d} t)^a$$, Appl. math. lett., 18, 7, 739-748, (2005) · Zbl 1082.60029 [6] Jumarie, G., Modified riemann – liouville derivative and fractional Taylor series of nondifferentiable functions further results, Comput. math. appl., 51, 9-10, 1367-1376, (2006) · Zbl 1137.65001 [7] Jumarie, G., Fractional partial differential equations and modified riemann – liouville derivative new methods for solution, J. appl. math. comput., 24, 1-2, 31-48, (2007) · Zbl 1145.26302 [8] Jumarie, G., Table of some basic fractional calculus formulae derived from a modified riemann – liouville derivative for non-differentiable functions, Appl. math. lett., 22, 3, 378-385, (2009) · Zbl 1171.26305 [9] Jumarie, G., Laplace’s transform of fractional order via the Mittag-Leffler function and modified riemann – liouville derivative, Appl. math. lett., 22, 11, 1659-1664, (2009) · Zbl 1181.44001 [10] Jumarie, G., Analysis of the equilibrium positions of nonlinear dynamical systems in the presence of coarse-graining disturbance in space, J. appl. math. comput., 32, 2, 329-351, (2010) · Zbl 1184.93067 [11] Jumarie, G., Cauchy’s integral formula via modified riemann – liouville derivative for analytic functions of fractional order, Appl. math. lett., 23, 12, 1444-1450, (2010) · Zbl 1202.30068 [12] Agrawal, O.P., Formulation of euler – lagrange equations for fractional variational problems, J. math. anal. appl., 272, 1, 368-379, (2002) · Zbl 1070.49013 [13] Almeida, R.; Torres, D.F.M., Calculus of variations with fractional derivatives and fractional integrals, Appl. math. lett., 22, 12, 1816-1820, (2009) · Zbl 1183.26005 [14] Almeida, R.; Torres, D.F.M., Leitmann’s direct method for fractional optimization problems, Appl. math. comput., 217, 3, 956-962, (2010) · Zbl 1200.65049 [15] Baleanu, D.; Muslih, S.I., Lagrangian formulation of classical fields within riemann – liouville fractional derivatives, Phys. scr., 72, 2-3, 119-121, (2005) · Zbl 1122.70360 [16] Frederico, G.S.F.; Torres, D.F.M., Fractional conservation laws in optimal control theory, Nonlinear dynam., 53, 3, 215-222, (2008) · Zbl 1170.49017 [17] Mozyrska, D.; Torres, D.F.M., Modified optimal energy and initial memory of fractional continuous-time linear systems, Signal process., 91, 3, 379-385, (2011) · Zbl 1203.94046 [18] Agrawal, O.P., Generalized euler – lagrange equations and transversality conditions for FVPs in terms of the Caputo derivative, J. vib. control, 13, 9-10, 1217-1237, (2007) · Zbl 1158.49006 [19] Almeida, R.; Torres, D.F.M., Necessary and sufficient conditions for the fractional calculus of variations with Caputo derivatives, Commun. nonlinear sci. numer. simul., 16, 3, 1490-1500, (2011) · Zbl 1221.49038 [20] Baleanu, D.; Agrawal, Om.P., Fractional Hamilton formalism within caputo’s derivative, Czech. J. phys., 56, 10-11, 1087-1092, (2006) · Zbl 1111.37304 [21] Frederico, G.S.F.; Torres, D.F.M., Fractional noether’s theorem in the riesz – caputo sense, Appl. math. comput., 217, 3, 1023-1033, (2010) · Zbl 1200.49019 [22] Malinowska, A.B.; Torres, D.F.M., Generalized natural boundary conditions for fractional variational problems in terms of the Caputo derivative, Comput. math. appl., 59, 9, 3110-3116, (2010) · Zbl 1193.49023 [23] Mozyrska, D.; Torres, D.F.M., Minimal modified energy control for fractional linear control systems with the Caputo derivative, Carpathian J. math., 26, 2, 210-221, (2010) · Zbl 1224.49048 [24] Bastos, N.R.O.; Ferreira, R.A.C.; Torres, D.F.M., Necessary optimality conditions for fractional difference problems of the calculus of variations, Discrete contin. dyn. syst., 29, 2, 417-437, (2011) · Zbl 1209.49020 [25] Bastos, N.R.O.; Ferreira, R.A.C.; Torres, D.F.M., Discrete-time fractional variational problems, Signal process., 91, 3, 513-524, (2011) · Zbl 1203.94022 [26] Cresson, J., Fractional embedding of differential operators and Lagrangian systems, J. math. phys., 48, 3, 033504, (2007), 34 pp · Zbl 1137.37322 [27] El-Nabulsi, R.A.; Torres, D.F.M., Fractional action-like variational problems, J. math. phys., 49, 5, 053521, (2008), 7 pp · Zbl 1152.81422 [28] Klimek, M., Lagrangian fractional mechanics—a noncommutative approach, Czech. J. phys., 55, 11, 1447-1453, (2005) [29] Atanacković, T.M.; Konjik, S.; Pilipović, S., Variational problems with fractional derivatives: euler – lagrange equations, J. phys. A, 41, 9, 095201, (2008), 12 pp · Zbl 1175.49020 [30] Jumarie, G., From Lagrangian mechanics fractal in space to space fractal schrödinger’s equation via fractional taylor’s series, Chaos solitons fractals, 41, 4, 1590-1604, (2009) · Zbl 1198.70019 [31] Almeida, R.; Malinowska, A.B.; Torres, D.F.M., A fractional calculus of variations for multiple integrals with application to vibrating string, J. math. phys., 51, 033503, (2010), 12 pp · Zbl 1309.49003 [32] Filatova, D.; Grzywaczewski, M.; Osmolovskii, N., Optimal control problem with an integral equation as the control object, Nonlinear anal., 72, 3-4, 1235-1246, (2010) · Zbl 1190.49033 [33] Jumarie, G., An approach via fractional analysis to non-linearity induced by coarse-graining in space, Nonlinear anal. RWA, 11, 1, 535-546, (2010) · Zbl 1195.37054 [34] Jumarie, G., Stock exchange fractional dynamics defined as fractional exponential growth driven by (usual) Gaussian white noise. application to fractional black – scholes equations, Insurance math. econom., 42, 1, 271-287, (2008) · Zbl 1141.91455 [35] Kolwankar, K.M., Decomposition of Lebesgue-Cantor devil’s staircase, Fractals, 12, 4, 375-380, (2004) · Zbl 1304.28007 [36] Jumarie, G., Modeling fractional stochastic systems as non-random fractional dynamics driven by Brownian motions, Appl. math. model., 32, 5, 836-859, (2008) · Zbl 1138.60324 [37] van Brunt, B., The calculus of variations, (2004), Springer New York · Zbl 1039.49001 [38] Almeida, R.; Torres, D.F.M., Hölderian variational problems subject to integral constraints, J. math. anal. appl., 359, 2, 674-681, (2009) · Zbl 1169.49016 [39] Almeida, R.; Torres, D.F.M., Isoperimetric problems on time scales with nabla derivatives, J. vib. control, 15, 6, 951-958, (2009) · Zbl 1273.49024 [40] Malinowska, A.B.; Torres, D.F.M., Necessary and sufficient conditions for local Pareto optimality on time scales, J. math. sci., 161, 6, 803-810, (2009) · Zbl 1192.49022 [41] Malinowska, A.B.; Sidi Ammi, M.R.; Torres, D.F.M., Composition functionals in fractional calculus of variations, Commun. frac. calc., 1, 1, 32-40, (2010)
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