This paper studies two-dimensional variable-order fractional optimal control problems (2D-VFOCPs) having dynamic constraints contain partial differential equations such as the convection–diffusion, diffusion-wave, and Burgers' equations. The variable-order time fractional derivative is described in the Caputo sense. To overcome computational difficulties, a novel numerical method based on transcendental Bernstein series (TBS) is proposed. In fact, we generalize the Bernstein polynomials to the larger class of functions which can provide more accurate approximate solutions. In this paper, we introduce the TBS and their properties, and subsequently, the privileges and effectiveness of these functions are demonstrated. Furthermore, we describe the approximation procedure which shows for solving 2D-VFOCPs how the needed basis functions can be determined. To do this, first we derive a number of new operational matrices of TBS. Second, the state and control functions are expanded in terms of the TBS with unknown free coefficients and control parameters. Then, based on these operational matrices and the Lagrange multipliers method, an optimization method is presented to an approximate solution of the state and control functions. Additionally, the convergence of the proposed method is analyzed. The results for several illustrative examples show that the proposed method is efficient and accurate.

References

References
1.
Coimbra
,
C. F. M.
,
2003
, “
Mechanica With Variable-Order Differential Operators
,”
Ann. Phys.
,
12
(
11–12
), pp.
692
703
.
2.
Cooper
,
G. R. J.
, and
Cowan
,
D. R.
,
2004
, “
Filtering Using Variable Order Vertical Derivatives
,”
Comput. Geosci.
,
30
(
5
), pp.
455
459
.
3.
Ingman
,
D.
, and
Suzdalnitsky
,
J.
,
2004
, “
Control of Damping Oscillations by Fractional Differential Operator With Time-Dependent Order
,”
Comput. Methods Appl. Mech. Eng.
,
193
(
52
), pp.
5585
5595
.
4.
Pedro
,
H. T. C.
,
Kobayashi
,
M. H.
,
Pereira
,
J. M. C.
, and
Coimbra
,
C. F. M.
,
2008
, “
Variable Order Modeling of Diffusive-Convective Effects on the Oscillatory Flow Past a Sphere
,”
J. Vib. Control
,
14
(
9–10
), pp.
1659
1672
.
5.
Tseng
,
C. C.
,
2006
, “
Design of Variable and Adaptive Fractional Order FIR Differentiators
,”
Signal Process.
,
86
(
10
), pp.
2554
2566
.
6.
Sun
,
H. G.
,
Chen
,
W.
, and
Chen
,
Y. Q.
,
2009
, “
Variable-Order Fractional Differential Operators in Anomalous Diffusion Modeling
,”
Physica A
,
388
(
21
), pp.
4586
4592
.
7.
Shen
,
S.
,
Liu
,
F.
,
Chen
,
J.
,
Turner
,
I.
, and
Anh
,
V.
,
2012
, “
Numerical Techniques for the Variable Order Time Fractional Diffusion Equation
,”
Appl. Math. Comput.
,
218
(22), pp.
10861
10870
.https://www.sciencedirect.com/science/article/abs/pii/S0096300312004468
8.
Dahaghin
,
M. S.
, and
Hassani
,
H.
,
2017
, “
An Optimization Method Based on the Generalized Polynomials for Nonlinear Variable-Order Time Fractional Diffusion-Wave Equation
,”
Nonlinear Dyn.
,
88
(
3
), pp.
1587
1598
.
9.
Bhrawy
,
A. H.
, and
Zaky
,
M. A.
,
2015
, “
Numerical Simulation for Two-Dimensional Variable-Order Fractional Nonlinear Cable Equation
,”
Nonlinear Dyn.
,
80
(
1–2
), pp.
101
116
.
10.
Yang
,
X. J.
, and
Machado
,
J. A.
T.,
2017
, “
A New Fractional Operator of Variable Order: Application in the Description of Anomalous Diffusion
,”
Physica A
,
481
, pp.
276
283
.
11.
Zhao
,
X.
,
Sun
,
Z. Z.
, and
Karniadakis
,
G. E.
,
2015
, “
Second-Order Approximations for Variable Order Fractional Derivatives: Algorithms and Applications
,”
J. Comput. Phys.
,
239
(15), pp.
184
200
.https://www.sciencedirect.com/science/article/pii/S0021999114005610
12.
Chen
,
Y.
,
Liu
,
L.
,
Li
,
B.
, and
Sun
,
Y.
,
2014
, “
Numerical Solution for the Variable Order Linear Cable Equation With Bernstein Polynomials
,”
Appl. Math. Comput.
,
238
(1), pp.
329
341
.https://www.sciencedirect.com/science/article/abs/pii/S0096300314004287
13.
Chen
,
Y. M.
,
Wei
,
Y. Q.
,
Liu
,
D. Y.
, and
Yu
,
H.
,
2015
, “
Numerical Solution for a Class of Nonlinear Variable Order Fractional Differential Equations With Legendre Wavelets
,”
Appl. Math. Lett.
,
46
, pp.
83
88
.
14.
Li
,
X.
,
Li
,
H.
, and
Wu
,
B.
,
2017
, “
A New Numerical Method for Variable Order Fractional Functional Differential Equations
,”
Appl. Math. Lett.
,
68
, pp.
80
86
.
15.
Li
,
X.
, and
Wu
,
B.
,
2015
, “
A Numerical Technique for Variable Fractional Functional Boundary Value Problems
,”
Appl. Math. Lett.
,
43
, pp.
108
113
.
16.
Zhang
,
H.
,
Liu
,
F.
,
Phanikumar
,
M. S.
, and
Meerschaert
,
M. M.
,
2013
, “
A Novel Numerical Method for the Time Variable Fractional Order Mobile-Immobile Advection-Dispersion Model
,”
Comput. Math. Appl.
,
66
(
5
), pp.
693
701
.
17.
Bohannan
,
G. W.
,
2008
, “
Analog Fractional Order Controller in Temperature and Motor Control Applications
,”
J. Vib. Control
,
14
(
9–10
), pp.
1487
1498
.
18.
Zamani
,
M.
,
Karimi-Ghartemani
,
M.
, and
Sadati
,
N.
,
2007
, “
FOPID Controller Design for Robust Performance Using Particle Swarm Optimization
,”
Fract. Calcul. Appl. Anal.
,
10
(
2
), pp.
169
187
.https://pdfs.semanticscholar.org/ad8f/94475e53c0b9da452fa6a1b16432a6bd94b0.pdf
19.
Tripathy
,
M. C.
,
Mondal
,
D.
,
Biswas
,
K.
, and
Sen
,
S.
,
2015
, “
Design and Performance Study of Phase-Locked Loop Using Fractional-Order Loop Filter
,”
Int. J. Circuit Theory Appl.
,
43
(
6
), pp.
776
792
.
20.
Khader
,
M. M.
, and
Hendy
,
A. S.
,
2012
, “
An Efficient Numerical Scheme for Solving Fractional Optimal Control Problems
,”
Int. J. Nonlinear Sci.
,
14
(
3
), pp.
287
296
.https://pdfs.semanticscholar.org/6f95/c5179f68c529b956c9721ff3c02066e66233.pdf
21.
Ezz-Eldien
,
S. S.
, and
El-Kalaawy
,
A. A.
,
2017
, “
Numerical Simulation and Convergence Analysis of Fractional Optimization Problems With Right-Sided Caputo Fractional Derivative
,”
ASME J. Comput. Nonlinear Dyn.
,
13
(
1
), p.
011010
.
22.
Effati
,
S.
,
Rakhshan
,
S. A.
, and
Saqi
,
S.
,
2018
, “
Formulation of Euler-Lagrange Equations for Multidelay Fractional Optimal Control Problems
,”
ASME J. Comput. Nonlinear Dyn.
,
13
(
6
), p.
061007
.
23.
Wang
,
B.
,
Xue
,
J.
,
Wu
,
F.
, and
Zhu
,
D.
,
2018
, “
Finite Time Takagi-Sugeno Fuzzy Control for Hydro-Turbine Governing System
,”
J. Vib. Control
,
24
(
5
), pp.
1001
1010
.
24.
Shi
,
K.
,
Wang
,
B.
, and
Chen
,
H.
,
2018
, “
Fuzzy Generalized Predictive Control for a Fractional-Order Nonlinear Hydro-Turbine Regulating System
,”
IET Renewable Power Gener.
,
12
(
14
), pp.
1708
1713
.
25.
Liu
,
L. Y.
,
Wang
,
B.
,
Wang
,
S.
,
Prey
,
S. E.
,
Hayat
,
T.
, and
Alsaadi
,
F. E.
,
2018
, “
Finite-Time H∞ Control of a Fractional-Order Hydraulic Turbine Governing System
,”
IEEE Access
,
6
, pp.
57507
57517
.
26.
Biswas
,
R. K.
, and
Sen
,
S.
,
2011
, “
Fractional Optimal Control Problems: A Pseudo-State-Space Approach
,”
J. Vib. Control
,
17
(
7
), pp.
1034
1041
.
27.
Jafari
,
H.
,
Ghasempour
,
S.
, and
Baleanu
,
D.
,
2016
, “
On Comparison Between Iterative Methods for Solving Nonlinear Optimal Control Problems
,”
J. Vib. Control
,
22
(
9
), pp.
2281
2287
.
28.
Rabiei
,
K.
,
Ordokhani
,
Y.
, and
Babolian
,
E.
,
2017
, “
The Boubaker Polynomials and Their Application to Solve Fractional Optimal Control Problems
,”
Nonlinear Dyn.
,
88
(
2
), pp.
1013
1026
.
29.
Lotfi
,
A.
,
Dehghan
,
M.
, and
Yousefi
,
S. A.
,
2011
, “
A Numerical Technique for Solving Fractional Optimal Control Problems
,”
Comput. Math. Appl.
,
62
(
3
), pp.
1055
1067
.
30.
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
.
31.
Nemati
,
A.
,
Yousefi
,
S.
,
Soltanian
,
F.
, and
Ardabili
,
J. S.
,
2016
, “
An Efficient Numerical Solution of Fractional Optimal Control Problems by Using the Ritz Method and Bernstein Operational Matrix
,”
Asian J. Control
,
18
(
6
), pp.
2272
2282
.
32.
Dehghan
,
M.
,
Hamedi
,
E. A.
, and
Khosravian-Arab
,
H.
,
2016
, “
A Numerical Scheme for the Solution of a Class of Fractional Variational and Optimal Control Problems Using the Modified Jacobi Polynomials
,”
J. Vib. Control
,
22
(
6
), pp.
1547
1559
.
33.
Ejlali
,
N.
, and
Hosseini
,
S. M.
,
2017
, “
A Pseudospectral Method for Fractional Optimal Control Problems
,”
J. Optim. Theory Appl.
,
174
(
1
), pp.
83
107
.
34.
Heydari
,
M. H.
,
Hooshmandasl
,
M. R.
,
Ghaini
,
F. M.
M., and
Cattani
,
C.
,
2016
, “
Wavelets Method for Solving Fractional Optimal Control Problems
,”
Appl. Math. Comput.
,
286
, pp.
139
154
.
35.
Almeida
,
R.
, and
Torres
,
D. F.
,
2015
, “
A Discrete Method to Solve Fractional Optimal Control Problems
,”
Nonlinear Dyn.
,
80
(
4
), pp.
1811
1816
.
36.
Bhrawy
,
A. H.
,
Doha
,
E. H.
,
Baleanu
,
D.
,
Ezz-Eldien
,
S. S.
, and
Abdelkawy
,
M. A.
,
2015
, “
An Accurate Numerical Technique for Solving Fractional Optimal Control Problems
,”
Proc. Rom. Acad., Ser. A
,
16
, pp.
47
54
.
37.
Tang
,
X.
,
Liu
,
Z.
, and
Wang
,
X.
,
2015
, “
Integral Fractional Pseudospectral Methods for Solving Fractional Optimal Control Problems
,”
Automatica
,
62
, pp.
304
311
.
38.
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
.
39.
Heydari
,
M. H.
, and
Avazzadeh
,
Z.
,
2018
, “
A New Wavelet Method for Variable-Order Fractional Optimal Control Problems
,”
Asian J. Control
,
20
(
5
), pp.
1804
1817
.
40.
Heydari
,
M. H.
, and
Avazzadeh
,
Z.
, “
A Computational Method for Solving Two-Dimensional Nonlinear Variable-Order Fractional Optimal Control Problems
,”
Asian J. Control
(in press).
41.
Tang
,
X.
,
Shi
,
Y.
, and
Wang
,
L. L.
,
2017
, “
A New Framework for Solving Fractional Optimal Control Problems Using Fractional Pseudospectral Methods
,”
Automatica
,
78
, pp.
333
340
.
42.
Wang
,
G.
,
Xiao
,
H.
, and
Xing
,
G.
,
2017
, “
An Optimal Control Problem for Mean-Field Forward-Backward Stochastic Differential Equation With Noisy Observation
,”
Automatica
,
86
, pp.
104
109
.
43.
Beuchler
,
S.
,
Hofer
,
K.
,
Wachsmuth
,
D.
, and
Wurst
,
J. E.
,
2015
, “
Boundary Concentrated Finite Elements for Optimal Control Problems With Distributed Observation
,”
Comput. Optim. Appl.
,
62
(
1
), pp.
31
65
.
44.
Tsai
,
J. S. H.
,
Li
,
J.
, and
Shieh
,
L. S.
,
2002
, “
Discretized Quadratic Optimal Control for Continuous-Time Two-Dimensional System
,”
IEEE Trans. Circuits Syst. I
,
49
(
1
), pp.
116
125
.
45.
Hasan
,
M. M.
,
Tangpong
,
X. W.
, and
Agrawal
,
O. P.
,
2012
, “
Fractional Optimal Control of Distributed Systems in Spherical and Cylindrical Coordinates
,”
J. Vib. Control
,
18
(
10
), pp.
1506
1525
.
46.
Özdemir
,
N.
,
Agrawal
,
O. P.
,
Iskender
,
B. B.
, and
Karadeniz
,
D.
,
2009
, “
Fractional Optimal Control of a 2-Dimensional Distributed System Using Eigenfunctions
,”
Nonlinear Dyn.
,
55
(
3
), pp.
251
260
.
47.
Nemati
,
A.
, and
Yousefi
,
S. A.
,
2017
, “
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.
,
34
(
4
), pp.
1079
1097
.https://academic.oup.com/imamci/article-abstract/34/4/1079/2669879?redirectedFrom=PDF
48.
Nemati
,
A.
,
2017
, “
Numerical Solution of 2D Fractional Optimal Control Problems by the Spectral Method Combined With Bernstein Operational Matrix
,”
Int. J. Control
,
91
(12), pp. 2632–2645.https://www.tandfonline.com/doi/abs/10.1080/00207179.2017.1334267
49.
Mamehrashi
,
K.
, and
Yousefi
,
S. A.
,
2017
, “
A Numerical Method for Solving a Nonlinear 2-D Optimal Control Problem With the Classical Diffusion Equation
,”
Int. J. Control
,
90
(
2
), pp.
298
306
.
50.
Rahimkhani
,
P.
, and
Ordokhani
,
Y.
,
2017
, “
Generalized Fractional-Order Bernoulli–Legendre Functions: An Effective Tool for Solving Two-Dimensional Fractional Optimal Control Problems
,”
IMA J. Math. Control Inf.
(in press).
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