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# Solving the Optimal Control Problems of Nonlinear Duffing Oscillators By Using an Iterative Shape Functions Method

Cheinshan Liu1, 2, Chunglun Kuo1, Jiangren Chang3, *

1 Center of Excellence for Ocean Engineering, National Taiwan Ocean University, Keelung, 20224, Taiwan.
2 Center of Excellence for the Oceans, National Taiwan Ocean University, Keelung, 20224, Taiwan.
3 Department of Systems Engineering and Naval Architecture, National Taiwan Ocean University, Keelung, 20224, Taiwan.

* Corresponding Author: Jiangren Chang. Email: .

(This article belongs to this Special Issue: Nonlinear Computational and Control Methods in Aerospace Engineering)

Computer Modeling in Engineering & Sciences 2020, 122(1), 33-48. https://doi.org/10.32604/cmes.2020.08490

## Abstract

In the optimal control problem of nonlinear dynamical system, the Hamiltonian formulation is useful and powerful to solve an optimal control force. However, the resulting Euler-Lagrange equations are not easy to solve, when the performance index is complicated, because one may encounter a two-point boundary value problem of nonlinear differential algebraic equations. To be a numerical method, it is hard to exactly preserve all the specified conditions, which might deteriorate the accuracy of numerical solution. With this in mind, we develop a novel algorithm to find the solution of the optimal control problem of nonlinear Duffing oscillator, which can exactly satisfy all the required conditions for the minimality of the performance index. A new idea of shape functions method (SFM) is introduced, from which we can transform the optimal control problems to the initial value problems for the new variables, whose initial values are given arbitrarily, and meanwhile the terminal values are determined iteratively. Numerical examples confirm the high-performance of the iterative algorithms based on the SFM, which are convergence fast, and also provide very accurate solutions. The new algorithm is robust, even large noise is imposed on the input data.

## Keywords

Liu, C., Kuo, C., Chang, J. (2020). Solving the Optimal Control Problems of Nonlinear Duffing Oscillators By Using an Iterative Shape Functions Method. CMES-Computer Modeling in Engineering & Sciences, 122(1), 33–48.

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