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Research Papers

Formulation of Euler–Lagrange Equations for Multidelay Fractional Optimal Control Problems

[+] Author and Article Information
Sohrab Effati

Department of Applied Mathematics,
Ferdowsi University of Mashhad,
Mashhad 91775-1159, Iran
e-mail: s-effati@um.ac.ir

Seyed Ali Rakhshan

Department of Applied Mathematics,
Ferdowsi University of Mashhad,
Mashhad 91775-1159, Iran
e-mail: seyedalirakhshan@yahoo.com

Samane Saqi

Department of Applied Mathematics,
Ferdowsi University of Mashhad,
Mashhad 91775-1159, Iran
e-mail: samane.saqi@mail.um.ac.ir

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received November 27, 2017; final manuscript received April 4, 2018; published online May 2, 2018. Assoc. Editor: Zaihua Wang.

J. Comput. Nonlinear Dynam 13(6), 061007 (May 02, 2018) (10 pages) Paper No: CND-17-1522; doi: 10.1115/1.4039900 History: Received November 27, 2017; Revised April 04, 2018

In this paper, a new numerical scheme is proposed for multidelay fractional order optimal control problems where its derivative is considered in the Grunwald–Letnikov sense. We develop generalized Euler–Lagrange equations that results from multidelay fractional optimal control problems (FOCP) with final terminal. These equations are created by using the calculus of variations and the formula for fractional integration by parts. The derived equations are then reduced into system of algebraic equations by using a Grunwald–Letnikov approximation for the fractional derivatives. Finally, for confirming the accuracy of the proposed approach, some illustrative numerical examples are solved.

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Topics: Optimal control
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Figures

Grahic Jump Location
Fig. 1

Approximation of state and control variable of Example 4.1 with different α: (a) state variable x(t) and (b) control variable u(t)

Grahic Jump Location
Fig. 2

Approximation of state and control variable of Example 4.2 with different α: (a) state variable x(t) and (b) control variable u(t)

Grahic Jump Location
Fig. 3

Approximation of state variable of Example 4.3 with α = 1: (a) state variable x1(t) and (b) state variable x2(t)

Grahic Jump Location
Fig. 4

Absolute error of control variable of Example 4.3 with α = 1

Grahic Jump Location
Fig. 5

Approximation of state variable of Example 4.4 with α = 1: (a) state variable x1(t) and (b) state variable x2(t)

Grahic Jump Location
Fig. 6

Absolute error of control variable of Example 4.4 with α = 1

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