Two schemes to find approximated solutions of optimal control problems of fractional order (FOCPs) are investigated. Integration and differentiation matrices were used in these schemes. These schemes used Chebyshev polynomials in the shifted case as a functional approximation. The target of the presented schemes is to convert such problems to optimization problems (OPs). Numerical examples are included, showing the strength of the schemes.
Abd ElalLFSweilamNHNagyAM, et al.(2016) Computational methods for the fractional optimal control HIV infection. Journal of Fractional Calculus and Applications7(2): 121–131.
2.
AgrawalOPBaleanuD (2007) A Hamiltonian formulation and a direct numerical scheme for fractional optimal control problems. Journal of Vibration and Control13(9–10): 1269–1281. DOI: 10.1177/1077546307077467.
3.
AhmedEMEl-SakaHA (2017) On a fractional order study of Middle East respiratory syndrome corona virus (MERS-COV). Journal of Fractional Calculus and Applications8(1): 118–126.
4.
AlizadehAEffatiSHeydariA (2017) Numerical schemes for fractional optimal control problems. Journal of Dynamic Systems, Measurement, and Control139(8): 081002DOI: 10.1115/1.4035533.
5.
AlmeidaRTorresDFM (2011) Necessary and sufficient conditions for the fractional calculus of variations with Caputo derivatives. Communications in Nonlinear Science and Numerical Simulation16(3): 1490–1500. DOI: 10.1016/j.cnsns.2010.07.016.
6.
AlmeidaRMalinowskaABTorresDFM (2010) A fractional calculus of variations for multiple integrals with application to vibrating string. Journal of Mathematical Physics51(3): 033503DOI: 10.1063/1.3319559.
7.
ArafaAAMHanafyIMGoudaMI (2016) Stability analysis of fractional order HIV infection of Cd4 + T cells with numerical solutions. Journal of Fractional Calculus and Applications7(1): 36–45.
8.
BehroozifarMHabibiN (2018) A numerical approach for solving a class of fractional optimal control problems via operational matrix Bernoulli polynomials. Journal of Vibration and Control24(12): 2494–2511. DOI: 10.1177/1077546316688608.
9.
BellWW (2004) Special Functions for Scientists and Engineers, Mineola, NY: Dover Publications, Inc .
10.
BhrawyAH (2016) A Jacobi spectral collocation method for solving multi-dimensional nonlinear fractional sub-diffusion equations. Numerical Algorithms73(1): 91–113. DOI: 10.1007/s11075-015-0087-2.
11.
BhrawyAHDohaEHBaleanuD, et al.(2015) An accurate numerical technique for solving fractional optimal control problems. Proceedings of the Romanian Academy, Series A16(1): 47–54.
12.
BhrawyAHDohaEHMachadoJAT, et al.(2015) An efficient numerical scheme for solving multi-dimensional fractional optimal control problems with a quadratic performance index. Asian Journal of Control17(6): 2389–2402. DOI: 10.1002/asjc.1109.
13.
BhrawyAHDohaEHEzz-EldienSS, et al.(2016) A numerical technique based on the shifted Legendre polynomials for solving the time-fractional coupled KdV equations. Calcolo53(1): 1–17. DOI: 10.1007/s10092-014-0132-x.
14.
DohaEHBhrawyAHEzz-EldienSS (2015) An efficient Legendre spectral tau matrix formulation for solving fractional subdiffusion and reaction subdiffusion equations. Journal of Computational and Nonlinear Dynamics10(2): 021019-1:021019–8. DOI: 10.1115/1.4027944.
15.
DumitruBKaiDEnricoS (2012) Fractional Calculus: Models and Numerical Methods, Boston, MA, USA: World Scientific .
El-GendiSE (1969) Chebyshev solution of differential, integral and integro-differential equations. Computer Journal12(3): 282–287. DOI: 10.1093/comjnl/12.3.282.
18.
El-KadyMEl-SayedA (2013) Fractional differentiation matrices for solving fractional orders differential equations. International Journal of Pure and Applied Mathematics84(2): 1–13. DOI: http://doi.org/10.12732/ijpam.v84i2.1.
19.
El-KadyMMoussaH (2011) Efficient monic Chebyshev pseudospectral method for solving integral and integro-differential equations. International Journal of Contemporary Mathematical Sciences6: 2261–2274.
20.
El-Kady M and Moussa H (2013) Monic Chebyshev approximations for solving optimal control problem with Volterra integro differential equations. 14(2): 23–26.
21.
El-KalaawyAADohaEHEzz-EldienSS, et al.(2018) A computationally efficient method for a class of fractional variational and optimal control problems using fractional Gegenbaue functions. Romanian Reports in Physics70(2): 90109 .
22.
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. DOI: 10.1177/1077546315573916.
23.
JumarieG (2007) Lagrangian mechanics of fractional order, Hamilton–Jacobi fractional PDE and Taylor's series of nondifferentiable functions. Chaos, Solitons & Fractals32(3): 969–987. DOI: 10.1016/j.chaos.2006.07.053.
24.
KeshavarzEOrdokhaniYRazzaghiM (2016) A numerical solution for fractional optimal control problems via Bernoulli polynomials. Journal of Vibration and Control22(18): 3889–3903. DOI: 10.1177/1077546314567181.
25.
KilbasAASrivastavaHMTrujilloJJ (2006) Theory and Applications of Fractional Differential Equations, Volume 204 (North-Holland Mathematics Studies), New York: Elsevier Science Inc.
26.
KiryakovaVS (1994) Generalized Fractional Calculus and Applications, New York: Longman Sci. Techn., Harlow, & John Wiley and Sons .
27.
LotfiAYousefiSADehghanM (2013) 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. Journal of Computational and Applied Mathematics250: 143–160. DOI: 10.1016/j.cam.2013.03.003.
28.
MainardiF (2012) An historical perspective on fractional calculus in linear viscoelasticity. Fractional Calculus and Applied Analysis15(4): 712–717. DOI: 10.2478/s13540-012-0048-6.
29.
MalinowskaABTorresDFM (2011) Fractional calculus of variations for a combined Caputo derivative. Fractional Calculus and Applied Analysis14(4): 523–2DOI: https://doi.org/10.2478/s13540-011-0032-6.
30.
MasonJCHandscombDC (2003) Chebyshev Polynomials, New York, NY: CRC, Boca Raton, Chapman and Hall .
31.
NematiAYousefiSA (2016) A numerical method for solving fractional optimal control problems using Ritz method. Journal of Computational and Nonlinear Dynamics11(5): 051015DOI: 10.1115/1.4032694.
32.
NematiAYousefiSSoltanianF, et 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. DOI: 10.1002/asjc.1321.
33.
OldhamKBSpanierJ (2006) The Fractional Calculus: Theory and Applications of Differentiation and Integration to Arbitrary Order, Mineola, New York: Dover Publications, Inc .
34.
PhangCIsmailNFIsahA, et al.(2018) A new efficient numerical scheme for solving fractional optimal control problems via a Genocchi operational matrix of integration. Journal of Vibration and Control24(14): 3036–3048. DOI: 10.1177/1077546317698909.
35.
SamkoSKilbasAAMarichevO (1993) Fractional Integrals and Derivatives: Theory and Applications, Yverdon: Gordon & Breach .
36.
ShenJTangTWangL-L (2011) Spectral Methods: Algorithms, Analysis and Applications, Heidelberg: Springer Science & Business Media .
37.
Tabatabaei SS, Yazdanpanah MJ and Tavazoei MS (2013) Incommensurate order fractional optimal control: Application to treatment of psychiatric disorders. In: 21st Iranian Conference on Electrical Engineering (ICEE), Mashhad, Iran, May 2013, pp. 1–5. DOI: 10.1109/IranianCEE.2013.6599831.
38.
TricaudCChenY (2010) An approximate method for numerically solving fractional order optimal control problems of general form. Computers & Mathematics with Applications59(5): 1644–1655. DOI: 10.1016/j.camwa.2009.08.006.
39.
YuanLKuangJ (2017) Stability and a numerical solution of fractional-order Brusselator chemical reaction system. Journal of Fractional Calculus and Applications8(1): 38–47.