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Research Papers

A Novel Method to Solve a Class of Distributed Optimal Control Problems Using Bezier Curves

[+] Author and Article Information
Majid Darehmiraki

Department of Applied Mathematics,
Ferdowsi University of Mashhad,
Mashhad 9177948974, Iran
e-mail: darehmiraki@gmail.com

Mohammad Hadi Farahi

Department of Applied Mathematics,
Ferdowsi University of Mashhad,
Mashhad 9177948974, Iran
e-mail: farahi@um.ac.ir

Sohrab Effati

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

1Corresponding author.

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received December 15, 2015; final manuscript received May 11, 2016; published online July 22, 2016. Assoc. Editor: Hiroshi Yabuno.

J. Comput. Nonlinear Dynam 11(6), 061008 (Jul 22, 2016) (13 pages) Paper No: CND-15-1439; doi: 10.1115/1.4033755 History: Received December 15, 2015; Revised May 11, 2016

In this paper, two approaches are used to solve a class of the distributed optimal control problems defined on rectangular domains. In the first approach, a meshless method for solving the distributed optimal control problems is proposed; this method is based on separable representation of state and control functions. The approximation process is done in two fundamental stages. First, the partial differential equation (PDE) constraint is transformed to an algebraic system by weighted residual method, and then, Bezier curves are used to approximate the action of control and state. In the second approach, the Bernstein polynomials together with Galerkin method are utilized to solve partial differential equation coupled system, which is a necessary and sufficient condition for the main problem. The proposed techniques are easy to implement, efficient, and yield accurate results. Numerical examples are provided to illustrate the flexibility and efficiency of the proposed method.

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Figures

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Fig. 1

Plots of target solution for Examples 8.1 (left) and 8.2 (right)

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Fig. 2

Graph objective function of Example 8.1 versus n for β=10−2,10−4, and 10−5

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Fig. 3

Graph ‖y−y¯‖L22 versus log(β) for Example 8.1 for RBF and Bezier methods

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Fig. 4

State variable in Example 8.1: β=10−5 (left) and β=10−2 (right)

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Fig. 5

Convergence of proposed method according to degree increasing of Bezier curves in Example 8.1

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Fig. 6

State and control function for Example 8.1 using second approach optimize-then-discretize for β=10−1, 10−2, and 10−5

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Fig. 7

Graph of the objective function associated with problem P versus log(β) for Example 8.1 with optimize-then-discretize approach and m = 6

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Fig. 8

Graph of the objective function associated with problem P and ‖y−y¯‖L22 versus degree of Bezier curves for Example 8.1 with optimize-then-discretize approach and β=10−6

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Fig. 9

Comparison of state variable with different degree of Bezier curves with n=20, β=10−3 for Example 8.2

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Fig. 10

Plots of the objective function associated with problem P for Example 8.2

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Fig. 11

Control and state functions for Example 8.2 using second approach optimize-then-discretize, from top to bottom, respectively, m=8, 10, and 12

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