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Research Papers

Numerical Schemes for Fractional Optimal Control Problems

[+] Author and Article Information
Ali Alizadeh

Department of Mathematics,
Payame Noor University,
P.O. Box 19395-3697,
Tehran 19395-3697, Iran
e-mail: Alizadeh312@gmail.com

Sohrab Effati

Department of Applied Mathematics,
Ferdowsi University of Mashhad;
Center of Excellent on Soft Computing and
Intelligent Information Processing,
Ferdowsi University of Mashhad,
Mashhad 9177948974, Iran
e-mail: s-effati@um.ac.ir

Aghileh Heydari

Department of Mathematics,
Payame Noor University,
P.O. Box 19395-3697,
Tehran 19395-3697, Iran
e-mail: a_heidari@pnu.ac.ir

1Corresponding author.

Contributed by the Dynamic Systems Division of ASME for publication in the JOURNAL OF DYNAMIC SYSTEMS, MEASUREMENT, AND CONTROL. Manuscript received May 22, 2016; final manuscript received December 10, 2016; published online May 15, 2017. Assoc. Editor: Ming Xin.

J. Dyn. Sys., Meas., Control 139(8), 081002 (May 15, 2017) Paper No: DS-16-1263; doi: 10.1115/1.4035533 History: Received May 22, 2016; Revised December 10, 2016

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.

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Figures

Grahic Jump Location
Fig. 1

For N = 10 and different values of α (–, exact; ○, α = 1; ⋆, α = 0.9; ♢, α = 0.7; and+, α = 0.5)

Grahic Jump Location
Fig. 2

For N = 10 and different values of α (–, exact; ○, α = 1; ⋆, α = 0.9; ♢, α = 0.7; and + , α = 0.5)

Grahic Jump Location
Fig. 3

For N = 10 and different values of α (–, exact; ⋆, α = 0.9; ○, α = 0.7; ♢, α = 0.5; and +, α = 0.3)

Grahic Jump Location
Fig. 4

Convergence of u(t) for α = 0.1 and different values of N (+, N = 20; Δ, N = 30; ○, N = 40; ⋆, N = 50; and , exact)

Grahic Jump Location
Fig. 5

Convergence of u(t) for α = 0.1 and different values of N (+, N = 20; Δ, N = 30; ○, N = 40; ⋆, N = 50; and , exact)

Grahic Jump Location
Fig. 6

For N = 10 and different values of α (⋆, α = 1(exact); ○, α = 1; +, α = 0.99; ♢, α = 0.9; and Δ, α = 0.8)

Grahic Jump Location
Fig. 7

For N = 10 and different values of α (⋆, α = 1(exact); ○, α = 1; +, α = 0.99; ♢, α = 0.9; and Δ, α = 0.8)

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