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In this paper, a pk-adaptive mesh refinement of pseudospectral method is proposed for solving optimal control problem by using collocation at Legendre-Gauss-Lobatto (LGL) points, motivated by reducing the redundant collocation points in the state-of-art mesh refinement methods to improve the time efficiency. The proposed method involves three phases, i.e., the determination of the polynomial degree, the determination of increasing intervals or nodes, and the optimization of the locations of segment breaks in each interval. First, determines the polynomial degree by the error estimation between the dynamics and the differentiation approximation of state variables according to the spectral matrix. Second, the maximum allowed polynomial degree in an interval is used to decide whether to segment interval or not. Third, the locations of segment points are obtained as the optimal design parameters of optimal control method. The terminology “pk-adaptive” or “p-then-k adaptive” is used because the polynomial degree is preferentially adaptive variation, then increases the segments by adding the optimal knots in each mesh interval. Finally, the residual of solutions, number of segments, number of nodes, CPU time, convergence of iteration, and parameters of the method have been analyzed in the comparing test to discuss the advantages of pk-adaptive mesh refinement. The discussions performed in two examples and demonstrated that the pk-adaptive method has the ability of optimizing nodes distribution to keep fewer nodes requirement and higher time efficiency than the hp- or ph-based pseudospectral methods while achieving the equivalent accuracy.

A pk-Adaptive Mesh Refinement for Pseudospectral Method to Solve Optimal Control Problem