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Article

Rabiei, Kobra ; Ordokhani, Yadollah
Boubaker hybrid functions and their application to solve fractional optimal control and fractional variational problems. (English). Applications of Mathematics, vol. 63 (2018), issue 5, pp. 541-567
MSC: 49J15, 49J40 | MR 3870148 | Zbl 06986925 | DOI: 10.21136/AM.2018.0083-18
Keywords:
fractional optimal control problems; fractional variational problems; Riemann-Liouville fractional integration; hybrid functions; Boubaker polynomials; Laplace transform; convergence analysis
Summary:
A new hybrid of block-pulse functions and Boubaker polynomials is constructed to solve the inequality constrained fractional optimal control problems (FOCPs) with quadratic performance index and fractional variational problems (FVPs). First, the general formulation of the Riemann-Liouville integral operator for Boubaker hybrid function is presented for the first time. Then it is applied to reduce the problems to optimization problems, which can be solved by the existing method. In this way we find the extremum value of FOCPs without adding slack variables to inequality trajectories. Also we show that if the number of bases is increased, the used approximations in this method are convergent. The applicability and validity of the method are shown by numerical results of some examples, moreover, a comparison with the existing results shows the preference of this method.
References:
[1] Agrawal, O. P.: 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 | MR 1930721 | Zbl 1070.49013
[2] Agrawal, O. P.: A general formulation and solution scheme for fractional optimal control problems. Nonlinear Dyn. 38 (2004), 323-337. DOI 10.1007/s11071-004-3764-6 | MR 2112177 | Zbl 1121.70019
[3] Agrawal, O. P.: A quadratic numerical scheme for fractional optimal control problems. J. Dyn. Syst. Meas. Control 130 (2007), 011010, 6 pages. DOI 10.1115/1.2814055 | MR 2463065
[4] Agrawal, O. P.: A general finite formulation for fractional variational problems. J. Math. Anal. Appl. 337 (2008), 1-12. DOI 10.1016/j.jmaa.2007.03.105 | MR 2356049 | Zbl 1123.65059
[5] Agrawal, O. P.: Fractional optimal control of a distributed system using eigenfunctions. J. Comput. Nonlinear Dyn. 3 (2008), 021204, 6 pages. DOI 10.1115/1.2833873 | MR 2466118
[6] Agrawal, O. P., Baleanu, D.: A Hamiltonian formulation and a direct numerical scheme for fractional optimal control problems. J. Vib. Control 13 (2007), 1269-1281. DOI 10.1177/1077546307077467 | MR 2356715 | Zbl 1182.70047
[7] Alipour, M., Rostamy, D., Baleanu, D.: Solving multi-dimensional fractional optimal control problems with inequality constraint by Bernstein polynomials operational matrices. J. Vib. Control 19 (2013), 2523-2540. DOI 10.1177/1077546312458308 | MR 3179222 | Zbl 1358.93097
[8] Almeida, R., Torres, D. F. M.: A discrete method to solve fractional optimal control problems. Nonlinear Dyn. 80 (2015), 1811-1816. DOI 10.1007/s11071-014-1378-1 | MR 3343434 | Zbl 1345.49022
[9] Bastos, N. R. O., Ferreira, R. A. C., Torres, D. F. M.: Discrete-time fractional variational problems. Signal Process. 91 (2011), 513-524. DOI 10.1016/j.sigpro.2010.05.001 | Zbl 1203.94022
[10] Bhrawy, A. H., Ezz-Eldien, S. S.: A new Legendre operational technique for delay fractional optimal control problems. Calcolo 53 (2016), 521-543. DOI 10.1007/s10092-015-0160-1 | MR 3574601 | Zbl 1377.49032
[11] Bryson, A. E., Ho, Y. C.: Applied Optimal Control. Optimization, Estimation, and Control. Hemisphere, Washington (1975). MR 0446628
[12] Canuto, C., Hussaini, M. Y., Quarteroni, A., Zang, T. A.: Spectral Methods. Fundamentals in Single Domains. Scientific Computation, Springer, Berlin (2006). DOI 10.1007/978-3-540-30726-6 | MR 2223552 | Zbl 1093.76002
[13] Darby, C. L., Hager, W. W., Rao, A. V.: An $hp$-adaptive pseudospectral method for solving optimal control problems. Optim. Control Appl. Methods 32 (2011), 476-502. DOI 10.1002/oca.957 | MR 2850736 | Zbl 1266.49066
[14] Dreyfus, S. F.: Variational problems with inequality constraints. J. Math. Anal. Appl. 4 (1962), 297-308. DOI 10.1016/0022-247X(62)90056-2 | MR 0141561 | Zbl 0119.16005
[15] El-Kady, M.: A Chebyshev finite difference method for solving a class of optimal control problems. Int. J. Comput. Math. 80 (2003), 883-895. DOI 10.1080/0020716031000070625 | MR 1984992 | Zbl 1037.65065
[16] Elnagar, G. N., Kazemi, M. A.: Pseudospectral Chebyshev optimal control of constrained nonlinear dynamical systems. Comput. Optim. Appl. 11 (1998), 195-217. DOI 10.1023/A:1018694111831 | MR 1652069 | Zbl 0914.93024
[17] Elnagar, G., Kazemi, M. A., Razzaghi, M.: The Pseudospectral Legendre method for discretizing optimal control problems. IEEE Trans. Autom. Control 40 (1995), 1793-1796. DOI 10.1109/9.467672 | MR 1354521 | Zbl 0863.49016
[18] Ezz-Eldien, S. S.: New quadrature approach based on operational matrix for solving a class of fractional variational problems. J. Comput. Phys. 317 (2016), 362-381. DOI 10.1016/j.jcp.2016.04.045 | MR 3498819 | Zbl 1349.65250
[19] Fahroo, F., Roos, I. M.: Direct trajectory optimization by a Chebyshev pseudospectral method. J. Guid Control Dyn. 25 (2002), 160-166. DOI 10.2514/2.4862
[20] Foroozandeh, Z., Shamsi, M.: Solution of nonlinear optimal control problems by the interpolating scaling functions. Acta Astronautica 72 (2012), 21-26. DOI 10.1016/j.actaastro.2011.10.004
[21] Gong, Q., Kang, W., Ross, I. M.: A pseudospectral method for the optimal control of constrained feedback linearizable systems. IEEE Trans. Autom. Control 51 (2006), 1115-1129. DOI 10.1109/TAC.2006.878570 | MR 2238794 | Zbl 1366.49035
[22] Holsapple, R., Venkataraman, R., Doman, D.: A modified simple shooting method for solving two-point boundary value problems. IEEE Aerospace Conf. Proc. 6 (2003), 2783-2790. DOI 10.1109/AERO.2003.1235204
[23] Jesus, I. S., Machado, J. A. Tenreiro: Fractional control of heat diffusion systems. Nonlinear Dyn. 54 (2008), 263-282. DOI 10.1007/s11071-007-9322-2 | MR 2442944 | Zbl 1210.80008
[24] Jiang, C., Lin, Q., Yu, C., Teo, K. L., Duan, G.-R.: An exact penalty method for free terminal time optimal control problem with continuous inequality constraints. J. Optim. Theory Appl. 154 (2012), 30-53. DOI 10.1007/s10957-012-0006-9 | MR 2931365 | Zbl 1264.49036
[25] Kafash, B., Delavarkhalafi, A., Karbassi, S. M., Boubaker, K.: A numerical approach for solving optimal control problems using the Boubaker polynomials expansion scheme. J. Interpolat. Approx. Sci. Comput. 2014 (2014), Article ID 00033, 18 pages. DOI 10.5899/2014/jiasc-00033 | MR 3200248
[26] Keshavarz, E., Ordokhani, Y., Razzaghi, Y.: A numerical solution for fractional optimal control problems via Bernoulli polynomials. J. Vib. Control 22 (2016), 3889-3903. DOI 10.1177/1077546314567181 | MR 3546331 | Zbl 1373.49003
[27] Khader, M. M.: An efficient approximate method for solving fractional variational problems. Appl. Math. Model. 39 (2015), 1643-1649. DOI 10.1016/j.apm.2014.09.012 | MR 3320820
[28] Khader, M. M., Hendy, A. S.: A numerical technique for solving fractional variational problems. Math. Methods Appl. Sci. 36 (2013), 1281-1289. DOI 10.1002/mma.2681 | MR 3072360 | Zbl 1281.65094
[29] Khalid, A., Huey, J., Singhose, W., Lawrence, J., Frakes, D.: Human operator performance testing using an input-shaped bridge crane. J. Dyn. Syst. Meas. Control 128 (2006), 835-841. DOI 10.1115/1.2361321
[30] Kilbas, A. A., Srivastava, H. M., Trujillo, J. J.: Theory and Applications of Fractional Differential Equations. North-Holland Mathematics Studies 204, Elsevier, Amsterdam (2006). DOI 10.1016/s0304-0208(06)x8001-5 | MR 2218073 | Zbl 1092.45003
[31] Kirk, D. E.: Optimal Control Theory: An Introduction. Dover Publication, New York (2004).
[32] Kreyszig, E.: Introductory Functional Analysis with Applications. John Wiley & Sons, New York (1978). MR 0467220 | Zbl 0368.46014
[33] Lancaster, P.: Theory of Matrices. Academic Press, New York (1969). MR 0245579 | Zbl 0186.05301
[34] Li, M., Peng, H.: Solutions of nonlinear constrained optimal control problems using quasilinearization and variational pseudospectral methods. ISA Trans. 62 (2016), 177-192. DOI 10.1016/j.isatra.2016.02.007
[35] Lotfi, A., Yousefi, S. A., Dehghan, M.: Numerical solution of a class of fractional optimal control problems via the Legendre orthonormal basis combined with the operational matrix and the Gauss quadrature rule. J. Comput. Appl. Math. 250 (2013), 143-160. DOI 10.1016/j.cam.2013.03.003 | MR 3044581 | Zbl 1286.49030
[36] Lu, Z.: New a posteriori $L^\infty(L^2)$ and $L^2(L^2)$-error estimates of mixed finite element methods for general nonlinear parabolic optimal control problems. Appl. Math., Praha 61 (2016), 135-163. DOI 10.1007/s10492-016-0126-x | MR 3470771 | Zbl 1389.49018
[37] 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 (2010), 3110-3116. DOI 10.1016/j.camwa.2010.02.032 | MR 2610543 | Zbl 1193.49023
[38] Marzban, H. R., Hoseini, S. M.: A composite Chebyshev finite difference method for nonlinear optimal control problems. Commun. Nonlinear Sci. Numer. Simul. 18 (2013), 1347-1361. DOI 10.1016/j.cnsns.2012.10.012 | MR 3016889 | Zbl 1282.65075
[39] Marzban, H. R., Razzaghi, M.: Hybrid functions approach for linearly constrained quadratic optimal control problems. Appl. Math. Modelling 27 (2003), 471-485. DOI 10.1016/S0307-904X(03)00050-7 | Zbl 1020.49025
[40] Marzban, H. R., Tabrizidooz, H. R., Razzaghi, M.: A composite collocation method for the nonlinear mixed Volterra-Fredholm-Hammerstein integral equations. Commun. Nonlinear Sci. Numer. Simul. 16 (2011), 1186-1194. DOI 10.1016/j.cnsns.2010.06.013 | MR 2736626 | Zbl 1221.65340
[41] Mashayekhi, S., Razzaghi, M.: An approximate method for solving fractional optimal control problems by hybrid functions. J. Vib. Control 24 (2018), 1621-1631. DOI 10.1177/1077546316665956 | MR 3785609
[42] Mehra, R. K., Davis, R. E.: A generalized gradient method for optimal control problems with inequality constraints and singular arcs. IEEE Trans. Autom. Control 17 (1972), 69-79. DOI 10.1109/TAC.1972.1099881 | Zbl 0268.49038
[43] Motta, M., Sartori, C.: The value function of an asymptotic exit-time optimal control problem. NoDEA, Nonlinear Differ. Equ. Appl. 22 (2015), 21-44. DOI 10.1007/s00030-014-0274-1 | MR 3311892 | Zbl 1311.49006
[44] Muslih, S. I., Baleanu, D.: Formulation of Hamiltonian equations for fractional variational problems. Czech J. Phys. 55 (2005), 633-642. DOI 10.1007/s10582-005-0067-1 | MR 216935 | Zbl 1181.70017
[45] Nemati, A., Yousefi, S., Soltanian, F., Ardabili, J. S.: An efficient numerical solution of fractional optimal control problems by using the Ritz method and Bernstein operational matrix. Asian J. Control 18 (2016), 2272-2282. DOI 10.1002/asjc.1321 | MR 3580387 | Zbl 1359.65100
[46] Ordokhani, Y., Rahimkhani, P.: A numerical technique for solving fractional variational problems by Müntz-Legendre polynomials. J. Appl. Math. Comput. 58 (2018), 75-94. DOI 10.1007/s12190-017-1134-z | MR 3847032 | Zbl 06943475
[47] Ordokhani, Y., Razzaghi, M.: Linear quadratic optimal control problems with inequality constraints via rationalized Haar functions. Dyn. Contin. Discrete Impuls. Syst., Ser. B, Appl. Algorithms 12 (2005), 761-773. MR 2179602 | Zbl 1081.49026
[48] Park, C., Scheeres, D. J.: Determination of optimal feedback terminal controllers for general boundary conditions using generating functions. Automatica 42 (2006), 869-875. DOI 10.1016/j.automatica.2006.01.015 | MR 2207828 | Zbl 1137.49020
[49] Pu, Y.-F., Siarry, P., Zhou, J.-L., Zhang, N.: A fractional partial differential equation based multiscale denoising model for texture image. Math. Methods Appl. Sci. 37 (2014), 1784-1806. DOI 10.1002/mma.2935 | MR 3231073 | Zbl 1301.35203
[50] Rabiei, K., Ordokhani, Y., Babolian, E.: The Boubaker polynomials and their application to solve fractional optimal control problems. Nonlinear Dyn. 88 (2017), 1013-1026. DOI 10.1007/s11071-016-3291-2 | MR 3628368 | Zbl 1380.49058
[51] Rabiei, K., Ordokhani, Y., Babolian, E.: Fractional-order Boubaker functions and their applications in solving delay fractional optimal control problems. J. Vib. Control 24 (2018), 3370-3383. DOI 10.1177/1077546317705041 | MR 3841934
[52] Razzaghi, M., Yousefi, S.: Legendre wavelets direct method for variational problems. Math. Comput. Simul. 53 (2000), 185-192. DOI 10.1016/S0378-4754(00)00170-1 | MR 1784947
[53] Riewe, F.: Nonconservative Lagrangian and Hamiltonian mechanics. Phys. Rev. E. 53 (1996), 1890-1899. DOI 10.1103/PhysRevE.53.1890 | MR 1401316
[54] Riewe, F.: Mechanics with fractional derivatives. Phys. Rev. E. 55 (1997), 3581-3592. DOI 10.1103/PhysRevE.55.3581 | MR 1438729
[55] Schiff, J. L.: The Laplace Transform: Theory and Applications. Undergraduate Texts in Mathematics, Springer, New York (1999). DOI 10.1007/978-0-387-22757-3 | MR 1716143 | Zbl 0934.44001
[56] Schittkowski, K.: NLPQL: A FORTRAN subroutine solving constrained nonlinear programming problems. Ann. Oper. Res. 5 (1986), 485-500. DOI 10.1007/BF02739235 | MR 0948031
[57] Suárez, I. J., Vinagre, B. M., Chen, Y. Q.: A fractional adaptation scheme for lateral control of an AGV. J. Vib. Control 14 (2008), 1499-1511. DOI 10.1177/1077546307087434 | MR 2463075 | Zbl 1229.70086
[58] Tohidi, E., Nik, H. Saberi: A Bessel collocation method for solving fractional optimal control problems. Appl. Math. Model. 39 (2015), 455-465. DOI 10.1016/j.apm.2014.06.003 | MR 3282588
[59] Tricaud, C., Chen, Y.: An approximate method for numerically solving fractional order optimal control problems of general form. Comput. Math. Appl. 59 (2010), 1644-1655. DOI 10.1016/j.camwa.2009.08.006 | MR 2595937 | Zbl 1189.49045
[60] Tuan, L. A., Lee, S. G.: Sliding mode controls of double-pendulum crane systems. J. Mech. Sci. Technol. 27 (2013), 1863-1873. DOI 10.1007/s12206-013-0437-8
[61] Vlassenbroeck, J.: A Chebyshev polynomial method for optimal control with state constraints. Automatica 24 (1988), 499-506. DOI 10.1016/0005-1098(88)90094-5 | MR 0956571 | Zbl 0647.49023
[62] Wang, X., Peng, H., Zhang, S., Chen, B., Zhong, W.: A symplectic pseudospectral method for nonlinear optimal control problems with inequality constraints. ISA Trans. 68 (2017), 335-352. DOI 10.1016/j.isatra.2017.02.018 | MR 3534963
[63] Wang, D., Xiao, A.: Fractional variational integrators for fractional variational problems. Commun. Nonlinear Sci. Numer. Simul. 17 (2012), 602-610. DOI 10.1016/j.cnsns.2011.06.028 | MR 2834419 | Zbl 1239.49028
[64] Yan, N.: Superconvergence analysis and a posteriori error estimation of a finite element method for an optimal control problem governed by integral equations. Appl. Math., Praha 54 (2003), 267-283. DOI 10.1007/s10492-009-0017-5 | MR 2530543 | Zbl 1212.65256
[65] Yonthanthum, W., Rattana, A., Razzaghi, M.: An approximate method for solving fractional optimal control problems by the hybrid of block-pulse functions and Taylor polynomials. Optim. Control Appl. Methods 39 (2018), 873-887. DOI 10.1002/oca.2383 | MR 3796971 | Zbl 06909040
[66] Yousefi, S. A., Dehghan, M., Lotfi, A.: Generalized Euler-Lagrange equations for fractional variational problems with free boundary conditions. Comput. Math. Appl. 62 (2011), 987-995. DOI 10.1016/j.camwa.2011.03.064 | MR 2824686 | Zbl 1228.49016
[67] Yousefi, S. A., Lotfi, A., Dehghan, M.: The use of Legendre multiwavelet collocation method for solving the fractional optimal control problems. J. Vib. Control 17 (2011), 2059-2065. DOI 10.1177/1077546311399950 | MR 2895863 | Zbl 1271.65105
[68] Zaky, M. A.: A Legendre collocation method for distributed-order fractional optimal control problems. Nonlinear Dyn. 91 (2018), 2667-2681. DOI 10.1007/s11071-017-4038-4 | Zbl 1392.35331
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