This study develops an efficient numerical method for optimal control problems governed by fractional Volterra integro-differential equations. A new type of polynomials orthogonal with respect to a discrete norm, namely discrete Hahn polynomials, is introduced and its properties investigated. Fractional operational matrices for the orthogonal polynomials are also derived. A direct numerical algorithm supported by the discrete Hahn polynomials and operational matrices is used to approximate solution of optimal control problems governed by fractional Volterra integro-differential equations. Several examples are analyzed and the results compared with those of other methods. The required CPU time assesses the computational cost and complexity of the proposed method.
AsliBHSFlusserJ (2017) New discrete orthogonal moments for signal analysis. Signal Processing141: 57–73.
2.
BanksHTBurnsJA (1978Hereditary control problems: numerical methods based on averaging approximations. SIAM Journal on Control and Optimization16(2): 169–208.
3.
BelbasSA (1999) Iterative schemes for optimal control of Volterra integral equations. Nonlinear Analysis: Theory, Methods & Applications37(1): 57–79.
4.
BelbasSA (2007) A new method for optimal control of volterra integral equations. Applied Mathematics and Computation189(2): 1902–1915.
5.
BhrawyAHZakyMAMachadoJAT (2017) Numerical solution of the two-sided space-time fractional telegraph equation via chebyshev tau approximation. Journal of Optimization Theory and Applications174(1): 321–341.
6.
ChenLAnJZhuangQ (2017) Direct solvers for the biharmonic eigenvalue problems using Legendre polynomials. Journal of Scientific Computing70(3): 1030–1041.
7.
ChouJHHorngIR (1987) Optimal control of deterministic systems described by integrodifferential equations via Chebyshev series. ASME J. Dyn. Syst. Measure. Contr. 109: 345–348.
8.
Ezz-EldienSDohaEBaleanuD, et al. (2017) A numerical approach based on Legendre orthonormal polynomials for numerical solutions of fractional optimal control problems. Journal of Vibration and Control23(1): 16–30.
9.
GoertzROffnerP (2016a) On Hahn polynomial expansion of a continuous function of bounded variation. PAMM16(1): 655–656.
10.
GoertzROffnerP (2016b) Spectral Accuracy for the Hahn Polynomials. ArXiv e-prints: arXiv:1609.07291.
11.
HahnW (1949) Uber orthogonalpolynome, die qdifferenzengleichungen gengen. Mathematische Nachrichten2(1–2): 4–34.
12.
HeydariMHAvazzadehZ (2018) A new wavelet method for variable-order fractional optimal control problems. Asian Journal of Control20(5): 1804–1817.
13.
JajarmiAHajipourM (2016) An efficient recursive shooting method for the optimal control of time-varying systems with state time-delay. Applied Mathematical Modelling40(4): 2756–2769.
14.
KhanduziREbrahimzadehAPeyghamiMR (2018) A modified teaching-learning-based optimization for optimal control of Volterra integral systems. Soft Computing22(17): 5889–5899.
15.
MaleknejadKEbrahimzadehA (2014) Optimal control of Volterra integro-differential systems based on Legendre wavelets and collocation method. International Journal of Mathematical and Computational Sciences8(7): 1040–1044.
16.
MashayekhiSOrdokhaniYRazzaghiM (2013) Hybrid functions approach for optimal control of systems described by integro-differential equations. Applied Mathematical Modelling37(5): 3355–3368.
17.
MohammadiFMoradiLBaleanuD, et al. (2018) A hybrid functions numerical scheme for fractional optimal control problems: application to nonanalytic dynamic systems. Journal of Vibration and Control24(21): 5030–5043.
18.
MoradiLMohammadiFBaleanuD (2019) A direct numerical solution of time-delay fractional optimal control problems by using Chelyshkov wavelets. Journal of Vibration and Control25(2): 310–324.
19.
MuthukumarPGanesh PriyaB (2017) Numerical solution of fractional delay differential equation by shifted Jacobi polynomials. International Journal of Computer Mathematics94(3): 471–492.
20.
OldhamKBSpanierJ (1974) The Fractional Calculus. New York: Academic Press.
21.
PalanisamyKRGanti PrasadaR (1983) Optimal control of linear systems with delays in state and control via Walsh functions. IEE Proceedings D (Control Theory and Applications)130 (6): 300–312.
22.
RamadanMARaslanKREl DanafTS, et al. (2017) An exponential Chebyshev second kind approximation for solving high-order ordinary differential equations in unbounded domains, with application to Dawson's integral. Journal of the Egyptian Mathematical Society25(2): 197–205.
23.
RazmjooyNRamezaniM (2017) Analytical solution for optimal control by the second kind Chebyshev polynomials expansion. Iranian Journal of Science and Technology, Transactions A: Science41(4): 1017–1026.
24.
SamkoSGKilbasAAMarichevOI (1993) Fractional Integrals and Derivatives. Yverdon-les-Bains, Switzerland: Gordon and Breach Science Publishers.
25.
ShahKKhalilHKhanRA (2017) A generalized scheme based on shifted Jacobi polynomials for numerical simulation of coupled systems of multi-term fractional-order partial differential equations. LMS Journal of Computation and Mathematics20(1): 11–29.
TohidiESamadiORN (2013) Optimal control of nonlinear Volterra integral equations via Legendre polynomials. IMA Journal of Mathematical Control and Information30(1): 67–83.
28.
WangXT (2007) Numerical solutions of optimal control for time delay systems by hybrid of block-pulse functions and Legendre polynomials. Applied Mathematics and Computation184(2): 849–856.
29.
XiuDKarniadakisGE (2002) The Wiener-Askey Polynomial Chaos for Stochastic Differential Equations. SIAM journal on Scientific Computing24(2): 619–644.
