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.

References

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
.
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