In this article, a class of fractional optimal control problems (FOCPs) are solved using a direct method. We present a new operational matrix of the fractional derivative in the sense of Caputo based on the B-spline functions. Then we reduce the solution of fractional optimal control problem to a nonlinear programming (NLP) one, where some existing well-developed algorithms may be applied. Numerical results demonstrate the efficiency of the presented technique.
AbramowitzMStegunIA (1964) Handbook of Mathematical Functions: With Formulas, Graphs, and Mathematical Tables. Vol. 55. New York, NY: Courier Corporation.
2.
AgrawalPEl-SayedAA (2018) Non-standard finite difference and Chebyshev collocation methods for solving fractional diffusion equation. Physica A: Statistical Mechanics and Its Applications500(C): 40–49.
3.
AgarwalPAkbarMNawazRJleliM (2020a) Solutions of system of Volterra integro-differential equations using optimal homotopy asymptotic method. Mathematical Methods in the Applied Sciences44: 1–11.
4.
AgarwalPBaltaevaUAlikulovY (2020b) Solvability of the boundary-value problem for a linear loaded integro-differential equation in an infinite three-dimensional domain. Chaos, Solitons & Fractals140(C): 110108.
5.
AgrawalPDefterliOBaleanuD (2010) Fractional optimal control problems with several state and control variables. Journal of Vibration and Control16(May): 1967–1976.
6.
AhmadWMEl-KhazaliR (2007) Fractional-order dynamical models of love. Chaos, Solitons and Fractals33(4): 1367–1375.
7.
AlderremyAASaadKMAgarwalP, et al. (2020) Certain new models of the multi space-fractional Gardner equation. Physica A: Statistical Mechanics and Its Applications545(C): 123806.
8.
AlizadehAEffatiS (2018) An iterative approach for solving fractional optimal control problems. JVC/Journal of Vibration and Control24(1): 18–36.
9.
AlmeidaRTorresDF (2015) A discrete method to solve fractional optimal control problems. Nonlinear Dynamics80(4): 1811–1816.
10.
AvrielM (1976) Nonlinear Programming: Analysis and Methods. New York: Dover Books on Computer Science.
11.
BhrawyAEzz-EldienSDohaE, et al. (2017) Solving fractional optimal control problems within a Chebyshev–Legendre operational technique. International Journal of Control90(6): 1230–1244.
12.
ChuiCK (2016) An Introduction to Wavelets. New York: Elsevier.
13.
DasS (2008) Functional Fractional Calculus for System Identification and Controls. New York: Springer.
14.
DenizFNAlagozBBTanN, et al. (2016) An integer order approximation method based on stability boundary locus for fractional order derivative/integrator operators. ISA Transactions62: 154–163.
15.
DevaneyR (2018) An Introduction to Chaotic Dynamical Systems. United States: CRC Press.
16.
Edrisi-TabrizYHeydariA (2014) Generalized B-spline functions method for solving optimal control problems. Computational Methods for Differential Equations2(4): 243–255.
17.
EdrisiTabriz YLakestaniM (2015) Direct solution of nonlinear constrained quadratic optimal control problems using B-spline functions. Kybernetika51(1): 81–98.
18.
Edrisi TabrizYLakestaniMHeydariA (2016) Two numerical methods for nonlinear constrained quadratic optimal control problems using linear B-spline functions. Iranian Journal of Numerical Analysis and Optimization6(2): 17–38.
19.
ElnagarGNKazemiMA (1998) Pseudospect al Chebyshev optimal control of constrained nonlinear dynamical systems. Computational Optimization and Applications11(2): 195–217.
20.
JiangYWangXWangY (2012) On a stochastic heat equation with first order fractional noises and applications to finance. Journal of Mathematical Analysis and Applications396(2): 656–669.
21.
GattoPHesthavenJS (2015) Numerical approximation of the fractional laplacian via hp-finite elements, with an application to image denoising. Journal of Scientific Computing65(1): 249–270.
22.
GillPMurrayW (1977) Linearly constrained problems including linear and quadratic programming. The State of the Art in Numerical Analysis: 313–363.
23.
GoswamJCChanAK (2011) Fundamentals of Wavelets: Theory, Algorithms, and Applications. Vol. 233. Hoboken, New Jersey: John Wiley and Sons.
24.
KilbasAASrivastavaHMTrujilloJJ (2003) Fractional differential equations: A emergent field in applied and mathematical sciences. Factorization, Singular Operators and Related Problems: 151–173.
25.
KreyszigE (1978) Introductory Functional Analysis with Applications. New York: Wiley.
26.
LakestaniMDehghanMIrandoust-PakchinJ (2012) The construction of operational matrix of fractional derivatives using b-spline functions. Communications in Nonlinear Science and Numerical Simulation17(3): 1149–1162.
27.
LakestaniMRazzaghiMDehghanM (2005) Solution of nonlinear fredholm-hammerstein integral equations by using semiorthogonal spline wavelets. Mathematical Problems in Engineering2005: 113–121.
28.
LakestaniMRazzaghiMDehghanM (2006) Semiorthogonal spline wavelets approximation for Fredholm integro-differential equations. Mathematical Problems in Engineering2006: 1–12.
29.
LancasterPTismenetskyM (1985) The Theory of Matrices: With Applications. New York: Elsevier.
30.
Le MehauteACrepyG (1983) Introduction to transfer and motion in fractal media: The geometry of kinetics. Solid State Ionics9(1): 17–30.
31.
LiCZengF (2015) Numerical Methods for Fractional Calculus. Vol. 24. New York: CRC Press.
32.
LotfiADehghanMYousefiSA (2011) A numerical technique for solving fractional optimal control problems. Computers & Mathematics with Applications62(3): 1055–1067.
33.
Morales-DelgadoVFGómez-AguilarJFSaadKM, et al. (2019) Analytic solution for oxygen diffusion from capillary to tissues involving external force effects: A fractional calculus approach. Physica A: Statistical Mechanics and its Applications523(C): 48–65.
34.
NematiAYousefiSASoltanianF, 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.
35.
OldhamKBSpanierJ (1970) The replacement of Fick’s laws by a formulation involving semidifferentiation. Journal of Electroanalytical Chemistry and Interfacial Electrochemistry26(2–3): 331–341.
36.
PiresEJSMachadoJATde Moura OliveiraPB (2003) Fractional order dynamics in a GA planner. Signal Processing83(11): 2377–2386.
37.
PodlubnyI (1998) Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications. San Diego: Elsevier.
38.
PoosehSAlmeidaRTorresDF (2014) Fractional order optimal control problems with free terminal time. Journal of Industrial & Management Optimization10(2): 363–381.
39.
RabieiKOrdokhaniYBabolianE (2016) The boubaker polynomials and their application to solve fractional optimal control problems. Nonlinear Dynamics88(2): 1013–1026.
40.
RivlinTJ (1981) An Introduction to the Approximation of Functions. WALTHAM, MASSACHUSETTS; TORONTO; LONDON: Courier Corporation.
41.
SalatiABShamsiMTorresDFM (2019) Direct transcription methods based on fractional integral approximation formulas for solving nonlinear fractional optimal control problems. Communications in Nonlinear Science and Numerical Simulation67: 334–350.
42.
SaxenaRKMathaiAMHauboldHJ (2004) On generalized fractional kinetic equations. Physica A: Statistical Mechanics and its Applications344(3–4): 657–664.
StoerJBulirschR (2013) Introduction to Numerical Analysis. Vol. 12. New York, Berlin, Heidelberg: Springer Science and Business Media.
45.
SunLChenL (2015) Free vibrations of a taut cable with a general viscoelastic damper modeled by fractional derivatives. Journal of Sound and Vibration335: 19–33.
46.
SweilamNHAl-AjamiTMHoppeRHW (2013) Numerical solution of some types of fractional optimal control problems. The Scientific World Journal2013: 306237.
