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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.

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Figures

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

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