In this paper, a numerical scheme based on hybrid Chelyshkov functions (HCFs) is presented to solve a class of fractional optimal control problems (FOCPs). To this end, by using the orthogonal Chelyshkov polynomials, the HCFs are constructed and a general formulation for their operational matrix of the fractional integration, in the Riemann–Liouville sense, is derived. This operational matrix together with HCFs are used to reduce the FOCP to a system of algebraic equations, which can be solved by any standard iterative algorithm. Moreover, the application of presented method to the problems with a nonanalytic dynamic system is investigated. Numerical results confirm that the proposed HCFs method can achieve spectral accuracy to approximate the solution of FOCPs.
AgrawalOP (2004) A general formulation and solution scheme for fractional optimal control problems. Nonlinear Dynamics38(1): 323–337.
2.
AkbarianTKeyanpourM (2013) A new approach to the numerical solution of fractional order optimal control problems. Applications and Applied Mathematics8(2): 523–534.
3.
AlipourMRostamyDBaleanuD (2013) Solving multi-dimensional fractional optimal control problems with inequality constraint by Bernstein polynomials operational matrices. Journal of Vibration and Control19(16): 2523–2540.
4.
AlmeidaRTorresDFM (2015) A discrete method to solve fractional optimal control problems. Nonlinear Dynamics80(4): 1811–1816.
5.
Bhrawy AH, Doha EH, Baleanu D, et al. (2015) An accurate numerical technique for solving fractional optimal control problems. Proceedings of the Romanian Academy Series A 16(1): 47–54.
6.
BiswasRKSenS (2011) Fractional optimal control problems: a pseudo-state-space approach. Journal of Vibration and Control17(7): 1034–1041.
7.
ChelyshkovVS (1994) A variant of spectral method in the theory of hydrodynamic stability. Hydromechanics68: 105–109.
8.
DabiriAButcherEA (2017a) Efficient modified Chebyshev differentiation matrices for fractional differential equations. Communications in Nonlinear Science and Numerical Simulation50: 284–310.
9.
DabiriAButcherEA (2017b) Stable fractional Chebyshev differentiation matrix for numerical solution of fractional differential equations. Nonlinear Dynamics90(1): 185–201.
10.
DabiriAButcherEANazariM (2017a) Coefficient of restitution in fractional viscoelastic compliant impacts using fractional Chebyshev collocation. Journal of Sound and Vibration388: 230–244.
11.
DabiriAButcherEAPoursinaMet al.(2017b) Optimal periodic-gain fractional delayed state feedback control for linear fractional periodic time-delayed systems. IEEE Transactions on Automatic Control. DOI: 10.1109/TAC.2017.2731522.
12.
DehghanMHamediEAKhosravian-ArabH (2016) A numerical scheme for the solution of a class of fractional variational and optimal control problems using the modified Jacobi polynomials. Journal of Vibration and Control22(6): 1547–1559.
13.
DohaEHBhrawyAHBaleanuDet al.(2015) An efficient numerical scheme based on the shifted orthonormal Jacobi polynomials for solving fractional optimal control problems. Advances in Difference Equations2015(1): 1–17.
14.
EjlaliNHosseiniSM (2015) A pseudospectral method for fractional optimal control problems. Journal of Optimization Theory and Applications174(1): 83–107.
15.
FornbergB (1998) A Practical Guide to Pseudospectral Methods (Volume 1), Cambridge, UK: Cambridge University Press.
16.
HeydariMHHooshmandaslMRGhainiFMet al.(2016) Wavelets method for solving fractional optimal control problems. Applied Mathematics and Computation286(3): 139–154.
17.
JafariHGhasempourSBaleanuD (2016) On comparison between iterative methods for solving nonlinear optimal control problems. Journal of Vibration and Control22(9): 2281–2287.
18.
JajarmiABaleanuD (2017) Suboptimal control of fractional-order dynamic systems with delay argument. Journal of Vibration and Control. 24(12): 2430–2446.
19.
KhaderMMHendyAS (2012) An efficient numerical scheme for solving fractional optimal control problems. International Journal of Nonlinear Science14(3): 87–296.
20.
KurkcuOKAslanESezerM (2016) A numerical approach with error estimation to solve general integro-differential-difference equations using Dickson polynomials. Applied Mathematics and Computation276(3): 324–339.
21.
LotfiADehghanMYousefiSA (2011) A numerical technique for solving fractional optimal control problems. Computers & Mathematics with Applications62(3): 1055–1067.
22.
MachadoJAT (2013) Optimal controllers with complex order derivatives. Journal of Optimization Theory and Applications156(1): 2–12.
23.
MachadoJATLopesAM (2017) Complex and fractional dynamics. Entropy19(2): 1–4.
24.
MaleknejadKKajaniMT (2004) Solving linear integro-differential equation system by Galerkin methods with hybrid functions. Applied Mathematics and Computation159(3): 603–612.
25.
MarzbanHRRazzaghiM (2003) Hybrid functions approach for linearly constrained quadratic optimal control problems. Applied Mathematical Modelling27(6): 471–485.
26.
MarzbanHRRazzaghiM (2006) Solution of multi-delay systems using hybrid of block-pulse functions and Taylor series. Journal of Sound and Vibration292(3): 954–963.
27.
MashayekhiSOrdokhaniYRazzaghiM (2013) A hybrid functions approach for the Duffing equation. Physica Scripta88(2): 1–8.
28.
MoghaddamBPAghiliA (2012) A numerical method for solving linear non-homogenous fractional ordinary differential equation. Applied Mathematics & Information Sciences6(3): 441–445.
29.
MoghaddamBPYaghoobiSMachadoJAT (2016) An extended predictor-corrector algorithm for variable-order fractional delay differential equations. Journal of Computational and Nonlinear Dynamics11(6): 1–7.
30.
MoghaddamBPMachadoJATBehforoozH (2017) An integro quadratic spline approach for a class of variable-order fractional initial value problems. Chaos, Solitons & Fractals102: 354–360.
31.
MohammadiFHosseiniMMMohyud-DinST (2011) Legendre wavelet Galerkin method for solving ordinary differential equations with non-analytic solution. International Journal of Systems Science42(4): 579–585.
32.
MollahasaniNMoghadamMMAfroozK (2016) A new treatment based on hybrid functions to the solution of telegraph equations of fractional order. Applied Mathematical Modelling40(4): 2804–2814.
33.
NematiAYousefiSSoltanianFet al.(2016) An efficient numerical solution of fractional optimal control problems by using the Ritz method and Bernstein operational matrix. Asian Journal of Control18(6): 2272–2282.
34.
OldhamKSpanierJ (1974) The Fractional Calculus: Theory and Applications of Differentiation and Integration to Arbitrary Order, Mineola, NY: Dover Publications.
35.
PodlubnyI (1999) Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of their Solution and some of their Applications, New York, NY: Academic Press.
36.
RabieiKOrdokhaniYBabolianE (2016) The Boubaker polynomials and their application to solve fractional optimal control problems. Nonlinear Dynamics88(2): 1013–1026.
37.
YaghoobiSMoghaddamBPIvazK (2016) An efficient cubic spline approximation for variable-order fractional differential equations with time delay. Nonlinear Dynamics87(2): 815–826.
