In the present study, variational iteration and Adomian decomposition methods (ADMs) are applied for solving a class of fractional optimal control problems (FOCPs). Also, a comparative study between these two methods is presented. The fractional derivative (FD) in these problems is in the Caputo sense. To solve the problem, first the necessary optimality conditions of FOCP are achieved for a linear tracking fractional optimal control problem, and then, these two methods are used to solve the resulting fractional differential equations (FDEs). It is shown that the modified Adomian decomposition method and variational iteration method (VIM) use the same iterative formula for solving linear and nonlinear FOCPs. The convergence of the modified Adomian decomposition method is analytically studied and to illustrate the validity and applicability of the methods, some examples are provided.
Issue Section:
Research Papers
References
1.
Nemati
, A.
, and Yousefi
, S. A.
, 2016
, “A Numerical Scheme for Solving Two-Dimensional Fractional Optimal Control Problems by the Ritz Method Combined With Fractional Operational Matrix
,” IMA J. Math. Control Inf.
(Online).2.
Tarasov
, V.
, 2010
, Fractional Dynamics: Application of Fractional Calculus to Dynamics of Particles, Fields and Media
, Springer-Verlag
, Berlin
.3.
Baleanu
, D.
, Machado
, J. A. T.
, and Luo
, A. C. J.
, 2012
, Fractional Dynamics and Control
, Springer-Verlag
, New York
.4.
Diethelm
, K.
, 2010
, The Analysis of Fractional Differential Equations
, Springer-Verlag
, Berlin
.5.
Kilbas
, A. A.
, Srivastava
, H. M.
, and Trujillo
, J. J.
, 2006
, “Theory and Applications of Fractional Differential Equations
,” North-Holland Mathematics Studies
, Elsevier Science B.V
, Amsterdam, The Netherlands
.6.
Machado
, J. T.
, Kiryakova
, V.
, and Mainardi
, F.
, 2011
, “Recent History of Fractional Calculus
,” Commun. Nonlinear Sci. Numer. Simul.
, 16
(3
), pp. 1140
–1153
.7.
Agrawal
, O. P.
, 2004
, “A General Formulation and Solution Scheme for Fractional Optimal Control Problems
,” Nonlinear Dyn.
, 38
(1
), pp. 323
–337
.8.
Almeida
, R.
, and Torres
, D. F. M.
, 2011
, “Necessary and Sufficient Conditions for the Fractional Calculus of Variations With Caputo Derivatives
,” Commun. Nonlinear Sci. Numer. Simul.
, 16
(3
), pp. 1490
–1500
.9.
Rakhshan
, S. A.
, Effati
, S.
, and Kamyad
, A. V.
, 2016
, “Comments on a Discrete Method to Solve Fractional Optimal Control Problems
,” Nonlinear Dyn.
(Online).10.
Rakhshan
, S. A.
, Effati
, S.
, and Kamyad
, A. V.
, 2016
, “Solving a Class of Fractional Optimal Control Problems by the Hamilton–Jacobi–Bellman Equation
,” J. Vib. Control
(Online).11.
Agrawal
, O. P.
, 2007
, “A Quadratic Numerical Scheme for Fractional Optimal Control Problems
,” ASME J. Dyn. Syst. Meas. Control
, 130
(1
), p. 011010
.12.
Alipour
, M.
, Rostamy
, D.
, and Baleanu
, D.
, 2013
, “Solving Multi-Dimensional Fractional Optimal Control Problems With Inequality Constraint by Bernstein Polynomials Operational Matrices
,” J. Vib. Control
, 19
(16
), pp. 2523
–2540
.13.
Baleanu
, D.
, Defterli
, O.
, and Agrawal
, O. P.
, 2009
, “A Central Difference Numerical Scheme for Fractional Optimal Control Problems
,” J. Vib. Control
, 15
(4), pp. 547
–597
.14.
Bhrawy
, A. H.
, Doha
, E. H.
, Machado
, J. A. T.
, and Ezz-Eldien
, S. S.
, 2015
, “An Efficient Numerical Scheme for Solving Multi-Dimensional Fractional Optimal Control Problems With a Quadratic Performance Index
,” Asian J. Control
, 17
(6
), pp. 2389
–2402
.15.
Lotfi
, A.
, Dehghan
, M.
, and Yousefi
, S. A.
, 2011
, “A Numerical Technique for Solving Fractional Optimal Control Problems
,” J. Comput. Appl. Math.
, 62
(3
), pp. 1055
–1067
.16.
Nemati
, A.
, and Yousefi
, S. A.
, 2016
, “A Numerical Method for Solving Fractional Optimal Control Problems Using Ritz Method
,” ASME J. Comput. Nonlinear Dyn.
, 11
(5
), p. 051015
.17.
Sabouri
, J.
, Effati
, S.
, and Pakdaman
, M.
, 2016
, “A Neural Network Approach for Solving a Class of Fractional Optimal Control Problems
,” Neural Process. Lett.
(Online).18.
Yousefi
, S. A.
, Lotf
, A.
, and Dehghan
, M.
, 2011
, “The Use of a Legendre Multiwavelet Collocation Method for Solving the Fractional Optimal Control Problems
,” J. Vib. Control
, 17
(13
), pp. 2059
–2065
.19.
Betts
, J.
, 1998
, “Survey of Numerical Methods for Trajectory Optimization
,” J. Guid. Control Dyn.
, 21
(2
), pp. 193
–207
.20.
Kirk
, D. E.
, 1970
, Optimal Control Theory: An Introduction
, Dover Publications
, Mineola, NY
.21.
Pinch
, E. R.
, 1993
, Optimal Control Theory: An Introduction
, Oxford University Press
, London
.22.
Stryk
, O. V.
, and Bulirsch
, R.
, 1992
, “Direct and Indirect Methods for Trajectory Optimization
,” Ann. Oper. Res.
, 37
(1
), pp. 357
–373
.23.
Liao
, S. J.
, 2004
, Beyond Perturbation: Introduction to Homotopy Analysis Method
, Chapman and Hall/CRC
, Boca Raton, FL
.24.
He
, J. H.
, 1999
, “Homotopy Perturbation Technique
,” Comput. Methods Appl. Mech. Eng.
, 178
(3
), pp. 257
–262
.25.
He
, J. H.
, 2000
, “A Coupling Method of a Homotopy Technique and a Perturbation Technique for Non-Linear Problems
,” Int. J. Non-Linear Mech.
, 35
(1
), pp. 37
–43
.26.
He
, J. H.
, 2000
, “Vim for Autonomous Ordinary Differential Systems
,” Appl. Math. Comput.
, 114
(2
), pp. 115
–123
.27.
Adomian
, G.
, 1994
, “Solving Frontier Problems of Physics: The Decomposition Method
,” Fundamental Theories of Physics
, Kluwer Academic Publishers
, Dordrecht, The Netherlands
.28.
Adomian
, G.
, 1989
, Nonlinear Stochastic Systems Theory and Applications to Physics
, Kluwer Academic Publishers
, Boston, MA
.29.
Adomian
, G.
, 1988
, “A Review of the Decomposition Method in Applied Mathematics
,” J. Math. Anal. Appl.
, 135
(2
), pp. 501
–544
.30.
Adomian
, G.
, 1991
, “Solving Frontier Problems Modelled by Nonlinear Partial Differential Equations
,” Comput. Math. Appl.
, 22
(8
), pp. 91
–94
.31.
Duan
, J. S.
, Rach
, R.
, Baleanu
, D.
, and Wazwaz
, A. M.
, 2012
, “A Review of the Adomian Decomposition Method and Its Applications to Fractional Differential Equations
,” Commun. Fractional Calculus
, 3
(2
), pp. 73
–99
.32.
Jafari
, H.
, and Daftardar-Gejji
, V.
, 2006
, “Revised Adomian Decomposition Method for Solving a System of Non-Linear Equations
,” Appl. Math. Comput.
, 175
(1
), pp. 1
–7
.33.
Jafari
, H.
, and Daftardar-Gejji
, V.
, 2006
, “Solving a System of Nonlinear Fractional Differential Equations Using Adomian Decomposition
,” J. Comput. Appl. Math.
, 196
(2
), pp. 644
–652
.34.
Safari
, M.
, and Danesh
, M.
, 2011
, “Application of Adomians Decomposition Method for the Analytical Solution of Space Fractional Diffusion Equation
,” Adv. Pure Math.
, 1
(6
), pp. 345
–350
.35.
Wazwaz
, A. M.
, 2001
, “The Numerical Solution of Sixth-Order Boundary Value Problems by the Modified Decomposition Method
,” Appl. Math. Comput.
, 118
(2
), pp. 311
–325
.36.
Wazwaz
, A. M.
, 1999
, “A Reliable Modification of Adomian Decomposition Method
,” Appl. Math. Comput.
, 102
(1
), pp. 77
–86
.37.
Wazwaz
, A. M.
, and El-Sayed
, S. M.
, 2001
, “A New Modification of the Adomian Decomposition Method for Linear and Nonlinear Operators
,” Appl. Math. Comput.
, 122
(3
), pp. 393
–405
.38.
Zhang
, X.
, 2005
, “A Modification of the Adomian Decomposition Method for a Class of Nonlinear Singular Boundary Value Problems
,” J. Comput. Appl. Math.
, 180
(2
), pp. 377
–389
.39.
Luo
, X. G.
, Wu
, Q. B.
, and Zhang
, B. Q.
, 2006
, “Revisit on Partial Solutions in the Adomian Decomposition Method: Solving Heat and Wave Equations
,” J. Math. Anal. Appl.
, 321
(1
), pp. 353
–363
.40.
Luo
, X. G.
, 2005
, “A Two-Step Adomian Decomposition Method
,” Appl. Math. Comput.
, 170
(1
), pp. 570
–583
.41.
Hasan
, Y. Q.
, and Zhu
, L. M.
, 2008
, “Modified Adomian Decomposition Method for Singular Initial Value Problems in the Second Order Ordinary Differential Equations
,” Surv. Math. Appl.
, 3
, pp. 183
–193
.42.
Hosseini
, M. M.
, and Nasabzadeh
, H.
, 2007
, “Modified Adomian Decomposition Method for Specific Second Order Ordinary Differential Equations
,” Appl. Math. Comput.
, 186
(1
), pp. 117
–123
.43.
Jin
, C.
, and Liu
, M.
, 2005
, “A New Modification of Adomian Decomposition Method for Solving a Kind of Evolution Equations
,” Math. Probl. Eng.
, 169
(2
), pp. 953
–962
.44.
He
, J. H.
, 1999
, “Vim—A Kind of Non-Linear Analytical Technique: Some Examples
,” Int. J. Non-Linear Mech.
, 34
(4
), pp. 699
–708
.45.
Merdan
, M.
, 2012
, “On the Solutions Fractional Riccati Differential Equation With Modified Riemann–Liouville Derivative
,” Int. J. Differ. Equations
, 2012
, p. 346089
.46.
Wu
, G. C.
, and Baleanu
, D.
, 2013
, “Vim for the Burgers’ Flow With Fractional Derivatives-New Lagrange Multipliers
,” Appl. Math. Modell.
, 37
(9
), pp. 6183
–6190
.47.
Mirhosseini-Alizamini
, S. M.
, Effati
, S.
, and Heydari
, A.
, 2015
, “An Iterative Method for Suboptimal Control of Linear Time-Delayed Systems
,” Syst. Control Lett.
, 82
, pp. 40
–50
.48.
Alizadeh
, A.
, and Effati
, S.
, 2016
, “An Iterative Approach for Solving Fractional Optimal Control Problems
,” J. Vib. Control
(Online).49.
Biswaz
, R. K.
, and Sen
, S.
, 2014
, “Indirect Solution for Optimal Control Problems With a Pure State Constraint
,” IFAC Proc.
, 47
(3
), pp. 2456
–2461
.50.
Pontryagin
, L. S.
, Boltyanskii
, V. G.
, Gamkrelidze
, R. V.
, and Mishchenko
, E. F.
, 1962
, The Mathematical Theory of Optimal Processes
, Wiley-Interscience
, New York
.51.
Abbaoui
, K.
, and Cherruault
, Y.
, 1994
, “Convergence of Adomian Method Applied to Nonlinear Equations
,” Math. Comput. Modell.
, 20
(9
), pp. 69
–73
.52.
Hosseini
, M. M.
, and Nasabzadeh
, H.
, 2006
, “On the Convergence of Adomian Decomposition Method
,” Appl. Math. Comput.
, 182
(1
), pp. 536
–543
.53.
Rajaram
, R.
, and Najafi
, M.
, 2009
, “Analytical Treatment and Convergence of the Adomian Decomposition Method for a System of Coupled Damped Wave Equations
,” Appl. Math. Comput.
, 212
(1
), pp. 72
–81
.54.
Jafari
, H.
, Ghasempoor
, S.
, and Khalique
, C. M.
, 2013
, “A Comparison Between Adomian’s Polynomials and He’s Polynomials for Nonlinear Functional Equations
,” Math. Probl. Eng.
, 2013
, p. 943232
.55.
Lotfi
, A.
, and Yousefi
, S. A.
, 2014
, “Epsilon–Ritz Method for Solving a Class of Fractional Constrained Optimization Problems
,” J. Optim. Theory Appl.
, 163
(3
), pp. 884
–899
.Copyright © 2017 by ASME
You do not currently have access to this content.
