# American Institute of Mathematical Sciences

## Generalized variational calculus in terms of multi-parameters involving Atangana-Baleanu's derivatives and application

 Department of Mathematics, College of Science, Taibah University, Al-Madinah Al-Munawarah, Saudi Arabia

* Permanent address: Department of Mathematics and Computer Science, Faculty of Science, Beni-Suef University, Beni-Suef, Egypt. Email: Bahaa_gm@yahoo.com

Received  March 2018 Revised  May 2018 Published  March 2019

Fund Project: The author is supported by Taibah University, Dean of Scientific Research

In this paper, the generalized variational calculus in terms of multi-parameters involving Atangana-Baleanu's Derivatives are discussed. We consider the Hilfers generalized fractional derivative that in sense Atangana-Baleanu derivatives. We develop integration by parts formulas for the generalized fractional derivatives which are key to developing fractional variational calculus. It is shown that many derivatives used recently and their variational formulations can be obtained by setting different parameters to different values. The fractional Euler-Lagrange equations of fractional Lagrangians for constrained systems contains a fractional Hilfer-Atangana-Baleanu's derivatives with multi parameters are investigated. We also define fractional generalized momenta and provide fractional Hamiltonian formulations in terms of the new generalized derivatives. An example is presented to show applications of the formulations presented here. Some possible extensions of this research are also discussed. We present a general formulation and a solution scheme for a class of Fractional Optimal Control Problems (FOCPs) for those systems. The performance index of a FOCP is considered as a function of both the state and the control variables, and the dynamic constraints are expressed by a set of FDEs. The calculus of variations, the Lagrange multiplier, and the formula for fractional integration by parts are used to obtain Euler-Lagrange equations for the FOCP.

Citation: G. M. Bahaa. Generalized variational calculus in terms of multi-parameters involving Atangana-Baleanu's derivatives and application. Discrete & Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2020027
##### References:
 [1] T. Abdeljawad and D. Baleanu, Discrete fractional differences with nonsingular discrete Mittag-Leffler kernels, Adv. Diff. Eq., 2016 (2016), Paper No. 232, 18 pp. doi: 10.1186/s13662-016-0949-5. Google Scholar [2] T. Abdeljawad and D. Baleanu, Integration by parts and its applications of a new nonlocal fractional derivative with Mittag-Leffler nonsingular kernel, J. Nonlinear Sci. Appl., 10 (2017), 1098-1107. doi: 10.22436/jnsa.010.03.20. Google Scholar [3] O. P. Agrawal, Formulation of Euler-Lagrange equations for fractional variational problems, Journal of Mathematical Analysis and Applications, 272 (2002), 368-379. doi: 10.1016/S0022-247X(02)00180-4. Google Scholar [4] O. P. Agrawal, A general formulation and solution scheme for fractional optimal control problems, Nonlinear Dynamics, 38 (2004), 323-337. doi: 10.1007/s11071-004-3764-6. Google Scholar [5] O. P. Agrawal and D. Baleanu, Hamiltonian formulation and direct numerical scheme for fractional optimal control problems, Journal of Vibration and Control, 13 (2007), 1269-1281. doi: 10.1177/1077546307077467. Google Scholar [6] R. P. Agarwal, S. Baghli and M. Benchohra, Controllability for semilinear functional and neutral functional evolution equations with infinite delay in Frchet spaces, Appl. Math. Optim., 60 (2009), 253-274. doi: 10.1007/s00245-009-9073-1. Google Scholar [7] B. S. T. Alkahtani, Chua's circuit model with Atangana–Baleanu derivative with fractional order, Chaos, Solitons, Fractals, 89 (2016), 547-551. Google Scholar [8] N. Al-Salti, E. Karimov and K. Sadarangani, On a differential equation with caputo-fabrizio fractional derivative of order $1 < \beta\leq2$ and application to Mass-Spring-Damper system, Progr. Fract. Differ. Appl., 2 (2016), 257-263. Google Scholar [9] A. Atangana, Non validity of index law in fractional calculus: A fractional differential operator with Markovian and non-Markovian properties, Physica A: Statistical Mechanics and its Applications, 505 (2018), 688-706. doi: 10.1016/j.physa.2018.03.056. Google Scholar [10] A. Atangana and D. Baleanu, New fractional derivatives with non-local and non-singular kernel: theory and application to heat transfer model, Thermal Science, 20 (2016), 763-769. Google Scholar [11] A. Atangana and J. F. Gomez Aguila, Decolonisation of fractional calculus rules: Breaking commutativity and associativity to capture more natural phenomena, Eur. Phys. J. Plus, 133 (2018), 166. doi: 10.1140/epjp/i2018-12021-3. Google Scholar [12] G. M. Bahaa, Fractional optimal control problem for variational inequalities with control constraints, IMA J. Math. Control and Inform., 35 (2018), 107-122. doi: 10.1093/imamci/dnw040. Google Scholar [13] G. M. Bahaa, Fractional optimal control problem for differential system with control constraints, Filomat, 30 (2016), 2177-2189. doi: 10.2298/FIL1608177B. Google Scholar [14] G. M. Bahaa, Fractional optimal control problem for infinite order system with control constraints, Adv. Diff. Eq., 2016 (2016), Paper No. 250, 16 pp. doi: 10.1186/s13662-016-0976-2. Google Scholar [15] G. M. Bahaa, Fractional optimal control problem for differential system with delay argument, Adv. Diff. Eq., 2017 (2017), Paper No. 69, 19 pp. doi: 10.1186/s13662-017-1121-6. Google Scholar [16] G. M. Bahaa, Fractional optimal control problem for variable-order differential systems, Fract. Calc. Appl. Anal., 20 (2017), 1447-1470. doi: 10.1515/fca-2017-0076. Google Scholar [17] D. Baleanu and S. I. Muslih, Lagrangian formulation on classical fields within Riemann-Liouville fractional derivatives, Phys. Scr., 72 (2005), 119-121. doi: 10.1238/Physica.Regular.072a00119. Google Scholar [18] D. Baleanu and T. Avkar, Lagrangian with linear velocities within Riemann-Liouville fractional derivatives, Nuovo Cimnto B, 119 (2004), 73-79. Google Scholar [19] D. Baleanu and A. Fernandez, On some new properties of fractional derivatives with Mittag-Leffler kernel, Communications in Nonlinear Science and Numerical Simulation, 59 (2018), 444–462, arXiv: 1712.01762v1, (2017). doi: 10.1016/j.cnsns.2017.12.003. Google Scholar [20] D. Baleanu and OM. P. Agrawal, Fractional Hamilton formalism within Caputo's derivative, Czechoslovak Journal of Physics, 56 (2000), 1087-1092. doi: 10.1007/s10582-006-0406-x. Google Scholar [21] D. Baleanu, A. Jajarmi and M. Hajipour, A new formulation of the fractional optimal control problems involving Mittag Leffler nonsingular kernel, J Optim. Theory. Appl., 175 (2017), 718-737. doi: 10.1007/s10957-017-1186-0. Google Scholar [22] M. Caputo and M. Fabrizio, A new definition of fractional derivative without singular kernel, Progr Fract. Differ. Appl., 1 (2015), 73-75. Google Scholar [23] J. D. Djida, A. Atangana and I. Area, Numerical Computation of a Fractional Derivative with Non-Local and Non-Singular Kernel, Math. Model. Nat. Phenom., 12 (2017), 4-13. doi: 10.1051/mmnp/201712302. Google Scholar [24] J. D. Djida, G. M. Mophou and I. Area, Optimal control of diffusion equation with fractional time derivative with nonlocal and nonsingular mittag-leffler kernel, Journal of Optimization Theory and Applications, (2018), 1–18, arXiv: 1711.09070. doi: 10.1007/s10957-018-1305-6. Google Scholar [25] A. M. A. El-Sayed, On the stochastic fractional calculus operators, Journal of Fractional Calculus and Applications, 6 (2015), 101-109. Google Scholar [26] K. Ervin Lenzi, A. Tateishi Angel and V. Haroldo Ribeiro, The role of fractional time-derivative operators on anomalous diffusion, Frontiers in Physics, 5 (2017), 1-9. Google Scholar [27] S. F. Frederico Gastao and F. M. Torres Delfim, Fractional optimal control in the sense of Caputo and the fractional Noethers theorem., Int. Math. Forum, 3 (2008), 479-493. Google Scholar [28] J. F. Gomez-Aguilar, Irving-Mullineux oscillator via fractional derivatives with Mittag Leffler kernel, Chaos Soliton. Fract., 95 (2017), 179-186. doi: 10.1016/j.chaos.2016.12.025. Google Scholar [29] J. F. Gomez-Aguilar, Space time fractional diffusion equation using a derivative with nonsingular and regular kernel, Phys. A, 465 (2017), 562-572. doi: 10.1016/j.physa.2016.08.072. Google Scholar [30] J. F. Gomez-Aguilar, A. Atangana and J. F. Morales-Delgado, Electrical circuits RC, LC, and RL described by Atangana-Baleanu fractional derivatives, Int. J. Circ. Theor. Appl., 45 (2017), 1514-1533. doi: 10.1002/cta.2348. Google Scholar [31] F. M. Hafez, A. M. A. El-Sayed and M. A. El-Tawil, On a stochastic fractional calculus, Fractional Calculus and Applied Analysis, 4 (2001), 81-90. Google Scholar [32] R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific Publ. Co., Singapore, 2000. doi: 10.1142/9789812817747. Google Scholar [33] R. Hilfer, Threefold introduction to fractional derivatives, Anomalous Transport: Foundations and Applications, (2008), 17–73. doi: 10.1002/9783527622979.ch2. Google Scholar [34] J. Hristov, Transient heat diffusion with a non-singular fading memory, Therm Sci., 20 (2016), 757-762. Google Scholar [35] F. Jarad, T. Maraba and D. Baleanu, Fractional variational optimal control problems with delayed arguments, Nonlinear Dyn., 62 (2010), 609-614. doi: 10.1007/s11071-010-9748-9. Google Scholar [36] F. Jarad, T. Maraba and D. Baleanu, Higher order fractional variational optimal control problems with delayed arguments, Applied Mathematics and Computation, 218 (2012), 9234-9240. doi: 10.1016/j.amc.2012.02.080. Google Scholar [37] A. A. Kilbas, M. Saigo and K. Saxena, Generalized Mittag-Leffler function and generalized fractional calculus operators, Int. Tran. Spec. Funct., 15 (2004), 31-49. doi: 10.1080/10652460310001600717. Google Scholar [38] A. M. Mathai and H. J. Haubold, Special Functions for Applied Scientists. New York: Springer, 2008. doi: 10.1007/978-0-387-75894-7. Google Scholar [39] G. M. Mophou, Optimal control of fractional diffusion equation, Computers and Mathematics with Applications, 61 (2011), 68-78. doi: 10.1016/j.camwa.2010.10.030. Google Scholar [40] G. M. Mophou, Optimal control of fractional diffusion equation with state constraints, Computers and Mathematics with Applications, 62 (2011), 1413-1426. doi: 10.1016/j.camwa.2011.04.044. Google Scholar [41] N. Ozdemir, D. Karadeniz and B. B. Iskender, Fractional optimal control problem of a distributed system in cylindrical coordinates, Phys. Lett. A, 373 (2009), 221-226. doi: 10.1016/j.physleta.2008.11.019. Google Scholar [42] F. Riewe, Nonconservative Lagrangian and Hamiltonian Mechanics, Phys. Rev. E, 53 (1996), 1890-1899. doi: 10.1103/PhysRevE.53.1890. Google Scholar [43] F. Riewe, Mechanics with fractional derivatives, Phys. Rev. E, 55 (1997), 3581-3592. doi: 10.1103/PhysRevE.55.3581. Google Scholar [44] N. A. Sheikh, F. Ali, M. Saqib, et al. Comparison and analysis of the Atangana-Baleanu and Caputo-Fabrizio fractional derivatives for generalized Casson fluid model with heat generation and chemical reaction, Results Phys., 7 (2017), 789–800. doi: 10.1016/j.rinp.2017.01.025. Google Scholar

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##### References:
 [1] T. Abdeljawad and D. Baleanu, Discrete fractional differences with nonsingular discrete Mittag-Leffler kernels, Adv. Diff. Eq., 2016 (2016), Paper No. 232, 18 pp. doi: 10.1186/s13662-016-0949-5. Google Scholar [2] T. Abdeljawad and D. Baleanu, Integration by parts and its applications of a new nonlocal fractional derivative with Mittag-Leffler nonsingular kernel, J. Nonlinear Sci. Appl., 10 (2017), 1098-1107. doi: 10.22436/jnsa.010.03.20. Google Scholar [3] O. P. Agrawal, Formulation of Euler-Lagrange equations for fractional variational problems, Journal of Mathematical Analysis and Applications, 272 (2002), 368-379. doi: 10.1016/S0022-247X(02)00180-4. Google Scholar [4] O. P. Agrawal, A general formulation and solution scheme for fractional optimal control problems, Nonlinear Dynamics, 38 (2004), 323-337. doi: 10.1007/s11071-004-3764-6. Google Scholar [5] O. P. Agrawal and D. Baleanu, Hamiltonian formulation and direct numerical scheme for fractional optimal control problems, Journal of Vibration and Control, 13 (2007), 1269-1281. doi: 10.1177/1077546307077467. Google Scholar [6] R. P. Agarwal, S. Baghli and M. Benchohra, Controllability for semilinear functional and neutral functional evolution equations with infinite delay in Frchet spaces, Appl. Math. Optim., 60 (2009), 253-274. doi: 10.1007/s00245-009-9073-1. Google Scholar [7] B. S. T. Alkahtani, Chua's circuit model with Atangana–Baleanu derivative with fractional order, Chaos, Solitons, Fractals, 89 (2016), 547-551. Google Scholar [8] N. Al-Salti, E. Karimov and K. Sadarangani, On a differential equation with caputo-fabrizio fractional derivative of order $1 < \beta\leq2$ and application to Mass-Spring-Damper system, Progr. Fract. Differ. Appl., 2 (2016), 257-263. Google Scholar [9] A. Atangana, Non validity of index law in fractional calculus: A fractional differential operator with Markovian and non-Markovian properties, Physica A: Statistical Mechanics and its Applications, 505 (2018), 688-706. doi: 10.1016/j.physa.2018.03.056. Google Scholar [10] A. Atangana and D. Baleanu, New fractional derivatives with non-local and non-singular kernel: theory and application to heat transfer model, Thermal Science, 20 (2016), 763-769. Google Scholar [11] A. Atangana and J. F. Gomez Aguila, Decolonisation of fractional calculus rules: Breaking commutativity and associativity to capture more natural phenomena, Eur. Phys. J. Plus, 133 (2018), 166. doi: 10.1140/epjp/i2018-12021-3. Google Scholar [12] G. M. Bahaa, Fractional optimal control problem for variational inequalities with control constraints, IMA J. Math. Control and Inform., 35 (2018), 107-122. doi: 10.1093/imamci/dnw040. Google Scholar [13] G. M. Bahaa, Fractional optimal control problem for differential system with control constraints, Filomat, 30 (2016), 2177-2189. doi: 10.2298/FIL1608177B. Google Scholar [14] G. M. Bahaa, Fractional optimal control problem for infinite order system with control constraints, Adv. Diff. Eq., 2016 (2016), Paper No. 250, 16 pp. doi: 10.1186/s13662-016-0976-2. Google Scholar [15] G. M. Bahaa, Fractional optimal control problem for differential system with delay argument, Adv. Diff. Eq., 2017 (2017), Paper No. 69, 19 pp. doi: 10.1186/s13662-017-1121-6. Google Scholar [16] G. M. Bahaa, Fractional optimal control problem for variable-order differential systems, Fract. Calc. Appl. Anal., 20 (2017), 1447-1470. doi: 10.1515/fca-2017-0076. Google Scholar [17] D. Baleanu and S. I. Muslih, Lagrangian formulation on classical fields within Riemann-Liouville fractional derivatives, Phys. Scr., 72 (2005), 119-121. doi: 10.1238/Physica.Regular.072a00119. Google Scholar [18] D. Baleanu and T. Avkar, Lagrangian with linear velocities within Riemann-Liouville fractional derivatives, Nuovo Cimnto B, 119 (2004), 73-79. Google Scholar [19] D. Baleanu and A. Fernandez, On some new properties of fractional derivatives with Mittag-Leffler kernel, Communications in Nonlinear Science and Numerical Simulation, 59 (2018), 444–462, arXiv: 1712.01762v1, (2017). doi: 10.1016/j.cnsns.2017.12.003. Google Scholar [20] D. Baleanu and OM. P. Agrawal, Fractional Hamilton formalism within Caputo's derivative, Czechoslovak Journal of Physics, 56 (2000), 1087-1092. doi: 10.1007/s10582-006-0406-x. Google Scholar [21] D. Baleanu, A. Jajarmi and M. Hajipour, A new formulation of the fractional optimal control problems involving Mittag Leffler nonsingular kernel, J Optim. Theory. Appl., 175 (2017), 718-737. doi: 10.1007/s10957-017-1186-0. Google Scholar [22] M. Caputo and M. Fabrizio, A new definition of fractional derivative without singular kernel, Progr Fract. Differ. Appl., 1 (2015), 73-75. Google Scholar [23] J. D. Djida, A. Atangana and I. Area, Numerical Computation of a Fractional Derivative with Non-Local and Non-Singular Kernel, Math. Model. Nat. Phenom., 12 (2017), 4-13. doi: 10.1051/mmnp/201712302. Google Scholar [24] J. D. Djida, G. M. Mophou and I. Area, Optimal control of diffusion equation with fractional time derivative with nonlocal and nonsingular mittag-leffler kernel, Journal of Optimization Theory and Applications, (2018), 1–18, arXiv: 1711.09070. doi: 10.1007/s10957-018-1305-6. Google Scholar [25] A. M. A. El-Sayed, On the stochastic fractional calculus operators, Journal of Fractional Calculus and Applications, 6 (2015), 101-109. Google Scholar [26] K. Ervin Lenzi, A. Tateishi Angel and V. Haroldo Ribeiro, The role of fractional time-derivative operators on anomalous diffusion, Frontiers in Physics, 5 (2017), 1-9. Google Scholar [27] S. F. Frederico Gastao and F. M. Torres Delfim, Fractional optimal control in the sense of Caputo and the fractional Noethers theorem., Int. Math. Forum, 3 (2008), 479-493. Google Scholar [28] J. F. Gomez-Aguilar, Irving-Mullineux oscillator via fractional derivatives with Mittag Leffler kernel, Chaos Soliton. Fract., 95 (2017), 179-186. doi: 10.1016/j.chaos.2016.12.025. Google Scholar [29] J. F. Gomez-Aguilar, Space time fractional diffusion equation using a derivative with nonsingular and regular kernel, Phys. A, 465 (2017), 562-572. doi: 10.1016/j.physa.2016.08.072. Google Scholar [30] J. F. Gomez-Aguilar, A. Atangana and J. F. Morales-Delgado, Electrical circuits RC, LC, and RL described by Atangana-Baleanu fractional derivatives, Int. J. Circ. Theor. Appl., 45 (2017), 1514-1533. doi: 10.1002/cta.2348. Google Scholar [31] F. M. Hafez, A. M. A. El-Sayed and M. A. El-Tawil, On a stochastic fractional calculus, Fractional Calculus and Applied Analysis, 4 (2001), 81-90. Google Scholar [32] R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific Publ. Co., Singapore, 2000. doi: 10.1142/9789812817747. Google Scholar [33] R. Hilfer, Threefold introduction to fractional derivatives, Anomalous Transport: Foundations and Applications, (2008), 17–73. doi: 10.1002/9783527622979.ch2. Google Scholar [34] J. Hristov, Transient heat diffusion with a non-singular fading memory, Therm Sci., 20 (2016), 757-762. Google Scholar [35] F. Jarad, T. Maraba and D. Baleanu, Fractional variational optimal control problems with delayed arguments, Nonlinear Dyn., 62 (2010), 609-614. doi: 10.1007/s11071-010-9748-9. Google Scholar [36] F. Jarad, T. Maraba and D. Baleanu, Higher order fractional variational optimal control problems with delayed arguments, Applied Mathematics and Computation, 218 (2012), 9234-9240. doi: 10.1016/j.amc.2012.02.080. Google Scholar [37] A. A. Kilbas, M. Saigo and K. Saxena, Generalized Mittag-Leffler function and generalized fractional calculus operators, Int. Tran. Spec. Funct., 15 (2004), 31-49. doi: 10.1080/10652460310001600717. Google Scholar [38] A. M. Mathai and H. J. Haubold, Special Functions for Applied Scientists. New York: Springer, 2008. doi: 10.1007/978-0-387-75894-7. Google Scholar [39] G. M. Mophou, Optimal control of fractional diffusion equation, Computers and Mathematics with Applications, 61 (2011), 68-78. doi: 10.1016/j.camwa.2010.10.030. Google Scholar [40] G. M. Mophou, Optimal control of fractional diffusion equation with state constraints, Computers and Mathematics with Applications, 62 (2011), 1413-1426. doi: 10.1016/j.camwa.2011.04.044. Google Scholar [41] N. Ozdemir, D. Karadeniz and B. B. Iskender, Fractional optimal control problem of a distributed system in cylindrical coordinates, Phys. Lett. A, 373 (2009), 221-226. doi: 10.1016/j.physleta.2008.11.019. Google Scholar [42] F. Riewe, Nonconservative Lagrangian and Hamiltonian Mechanics, Phys. Rev. E, 53 (1996), 1890-1899. doi: 10.1103/PhysRevE.53.1890. Google Scholar [43] F. Riewe, Mechanics with fractional derivatives, Phys. Rev. E, 55 (1997), 3581-3592. doi: 10.1103/PhysRevE.55.3581. Google Scholar [44] N. A. Sheikh, F. Ali, M. Saqib, et al. Comparison and analysis of the Atangana-Baleanu and Caputo-Fabrizio fractional derivatives for generalized Casson fluid model with heat generation and chemical reaction, Results Phys., 7 (2017), 789–800. doi: 10.1016/j.rinp.2017.01.025. Google Scholar
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