Research Papers

Solving Two-Dimensional Variable-Order Fractional Optimal Control Problems With Transcendental Bernstein Series

[+] Author and Article Information
Hossein Hassani

Department of Applied Mathematics,
Faculty of Mathematical Sciences,
Shahrekord University,
Shahrekord 115, Iran
e-mail: hosseinhassani40@yahoo.com

Zakieh Avazzadeh

School of Mathematical Sciences,
Nanjing Normal University,
Nanjing 210023, China
e-mail: z.avazzadeh@njnu.edu.cn

José António Tenreiro Machado

Department of Electrical Engineering,
Institute of Engineering,
Polytechnic of Porto,
R. Dr. António Bernardino de Almeida,
Porto 431 4249-015, Portugal
e-mail: jtm@isep.ipp.pt

1Corresponding author.

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received April 23, 2018; final manuscript received February 19, 2019; published online April 8, 2019. Assoc. Editor: Bernard Brogliato.

J. Comput. Nonlinear Dynam 14(6), 061001 (Apr 08, 2019) (11 pages) Paper No: CND-18-1182; doi: 10.1115/1.4042997 History: Received April 23, 2018; Revised February 19, 2019

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.

Copyright © 2019 by ASME
Your Session has timed out. Please sign back in to continue.


Coimbra, C. F. M. , 2003, “ Mechanica With Variable-Order Differential Operators,” Ann. Phys., 12(11–12), pp. 692–703. [CrossRef]
Cooper, G. R. J. , and Cowan, D. R. , 2004, “ Filtering Using Variable Order Vertical Derivatives,” Comput. Geosci., 30(5), pp. 455–459. [CrossRef]
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. [CrossRef]
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. [CrossRef]
Tseng, C. C. , 2006, “ Design of Variable and Adaptive Fractional Order FIR Differentiators,” Signal Process., 86(10), pp. 2554–2566. [CrossRef]
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. [CrossRef]
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
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. [CrossRef]
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. [CrossRef]
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. [CrossRef]
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
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
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. [CrossRef]
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. [CrossRef]
Li, X. , and Wu, B. , 2015, “ A Numerical Technique for Variable Fractional Functional Boundary Value Problems,” Appl. Math. Lett., 43, pp. 108–113. [CrossRef]
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. [CrossRef]
Bohannan, G. W. , 2008, “ Analog Fractional Order Controller in Temperature and Motor Control Applications,” J. Vib. Control, 14(9–10), pp. 1487–1498. [CrossRef]
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
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. [CrossRef]
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
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. [CrossRef]
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. [CrossRef]
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. [CrossRef]
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. [CrossRef]
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. [CrossRef]
Biswas, R. K. , and Sen, S. , 2011, “ Fractional Optimal Control Problems: A Pseudo-State-Space Approach,” J. Vib. Control, 17(7), pp. 1034–1041. [CrossRef]
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. [CrossRef]
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. [CrossRef]
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. [CrossRef]
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. [CrossRef]
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. [CrossRef]
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. [CrossRef]
Ejlali, N. , and Hosseini, S. M. , 2017, “ A Pseudospectral Method for Fractional Optimal Control Problems,” J. Optim. Theory Appl., 174(1), pp. 83–107. [CrossRef]
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.
Almeida, R. , and Torres, D. F. , 2015, “ A Discrete Method to Solve Fractional Optimal Control Problems,” Nonlinear Dyn., 80(4), pp. 1811–1816. [CrossRef]
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.
Tang, X. , Liu, Z. , and Wang, X. , 2015, “ Integral Fractional Pseudospectral Methods for Solving Fractional Optimal Control Problems,” Automatica, 62, pp. 304–311. [CrossRef]
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. [CrossRef]
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. [CrossRef]
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).
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. [CrossRef]
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. [CrossRef]
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. [CrossRef]
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. [CrossRef]
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. [CrossRef]
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. [CrossRef]
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
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
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. [CrossRef]
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).


Grahic Jump Location
Fig. 3

The approximate state function z(x, t) (left side) and control function u(x, t) (right side) for α(x,t)=1−0.4 e−(xt)2−1 in example 4 with m1=3, m2=2, n1=3, and n2=3

Grahic Jump Location
Fig. 1

The approximate state function z(x, t) (left side) and control function u(x, t) (right side) for α(x,t)=1−0.5 cos(xt) in example 2 where m1=3, m2=2, n1=2, and n2=2

Grahic Jump Location
Fig. 2

The approximate state function z(x, t) (left side) and control function u(x, t) (right side) for α(x,t)=0.2+0.3(x2+t2) for example 3 with m1=4, m2=2, n1=4, and n2=2



Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content
Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In